Because of polymath10, I did not discuss over here other things. Let me mention two super major developments that I am sure you all know about. One is Laci Babai’s quasi-polynomial algorithm for Graph isomorphism. (This is a good time to mention that my wife’s mother’s maiden name is Babai.) You can read about it here (and the next three posts) and here. Another is the solution by Jean Bourgain, Ciprian Demeter, Larry Guth of Vinogradov’s main conjecture. You can read about it here and here.

We denoted by the largest cardinality of a family of -sets without a sunflower of size , and the largest cardinality in a family of -subsets of with the property , namely without a sunflower of size with head of size at most . Adding the subscript for refer to balanced families rather than to arbitrary family. The “Erdos-Ko-Rado regime” is when and is arbitrary or when and is arbitrary. (In this regime the property is preserved under shifting.)

Ferdinand gave interesting upper bounds and lower bounds for in terms of .

For the upper bound see this writeup by Ferdinand. For more details see this comment and the following ones. It will be interesting to find sharp lower and upper bounds.

Domotor raised in this comment and the following ones some ideas for using algebraic methods for the sunflower conjecture. This have led to an interesting discussion (following these comments) between Domotor and Ferdinand on connection between Sunflowers and Ramsey numbers (mainly ).

So that this post will not interrupt this interesting discussion let me quote the beginning of Domotor’s comment posted minutes ago:

“So let us denote the size of the largest -uniform family without (different) sets whose pairwise intersection has the same size by . As we have seen in the earlier discussion, trivially and (where in the last Ramsey notation we forbid in a -coloring of the complete graph). Mysteriously, for all three functions we have similar lower and upper bounds. In this comment I propose to try proving for some .”

Shachar Lovett proposed in this comment how to use the topological Tverberg theorem for attacking a certain important special case of the conjecture.

We started with the observation that every family of -sets contains a balanced subfamily of size . (BTW for graphs there is a sharper result by Noga Alon and it would be interesting to extend it to hypergraphs!) This gives that

Phillip Gibbs showed in this comment that if the sunflower conjecture is true then

It will be interesting to describe precisely what (1+o(1)) stands for, and improve it as much as possible. All left to be done for a counterexample to the sunflower conjecture is to complement Phillip construction by another one showing an exponential gap between the balanced and general case. This is certainly a tempting direction.

More drastic reductions then the very basic one are also possible, for example: for families of -subsets without a sunflower of size 3 it is enough to consider families with elements colored by colors such that every set is colored by consecutive colors and two sets sharing elements are not using the same color-intervals. (This implies “high girth” of some sort.) We can ask if this type of reductions is useful for proving the conjecture, if one can show (like Phillip) that assuming the conjecture, the gap between the bounds for this version is not too large compared to the general version, and hope that this can be part of a construction going in the negative direction.

In quite a few comments to post 3, I tried to further develop the homological ideas (and to mention some low hanging fruits and related questions). We did reach a major obstacle sending us back to the drawing board. I see some possible way around the difficulty and I will now describe it. For this purpose I will review the homological ideas in somewhat more general and very elementary context. (We only briefly mention the connection with cycles/homology but it is not needed).

Let be a generic matrix. The -th compound matrix is the matrix of by minors. Namely, , where .

Given two -unform hypergraphs we say that and are **weakly isomorphic** if the minor of whose rows and columns correspond to sets in and respectively is non-singular. (It is fairly easy to see that if and are isomorphic then they are weakly isomorphic. This relies on the condition that is generic.) We will say that **dominates** if and is full rank. Thus, and are weakly isomorphic if each dominates the other.

Let be the set of -subsets of such that . The connection with our notions of acyclicity is as follows: is acyclic (i.e. ) iff is dominated by . In greater generality, iff is dominated by . (In particular, iff is dominated by , and iff is dominated by $D[m,m]$.) **Remark:** Here and later in the post with since we deal only with cycles for the top dimension, so there are no “boundaries” to mod out.

A family of -subsets of is **shifted** if whenever and is obtained by replacing an element by a smaller element then . It can be shown that two shifted families of the same size are weakly isomorphic only if they are equal! We can use our compound matrix to describe an operation (in fact, several operations) called shifting which associated to a family a shifted family . Suppose that . is the lexicographically smallest family of sets which is weakly isomorphic to . In more details: we first consider a total ordering on -subsets of . Then we greedily build a hypergraph which is dominated by . When we reach sets we obtain a hypergraph weakly isomorphic to .

Now, if the total ordering is the lexicographic order on then we denote and call the “algebraic shifting” of . In this case, it can be shown that the resulting family is shifted. Also of special importance is the case that is *the reverse lexicographic order*.

For sets of integers of size , the lexicographic order refers to the lexicographic order when we ordered the elements of every set from increasingly, and the reverse lexicographic order is the lexicographic order when we order the elements decreasingly.

From 12<13<14<15<…<21<22<…<31<32<… we get

and from 21<31<32<41<42<43<51<52<… we get

We mention again some connection with acyclicity and homology: is acyclic if all sets in contains ‘1’. is -acyclic iff all sets in in intersect . is -acyclic iff all sets in contains . For general , -acyclicity is not expressed by those two versions of algebraic shifting, however, is -cyclic if .

The following properties are preserved under algebraic shifting:

(1) Every two members of has at least elements in common.

(2) There are no pairwise disjoint sets in .

For balanced families we know even more:

(3) If is balanced and every two members of has at least elements in common then all sets in contains .

and

(4) If is balanced and there are no pairwise disjoint sets in , then every set in intersects .

As it turns out we have much stronger statements:

The following properties are preserved under reverse-lexicographic algebraic shifting:

(5) Every two members of has at least elements in common.

(6) There are no pairwise disjoint sets in .

(I dont know if (3) and (4) extends as well. It certainly worth the effort to check it.)

Our ultimate conjecture remains the same:

**Main Conjecture:** If has no sunflower of size then it is -acyclic. (I.e., )

For balanced family we conjectured that and for the non-balance we had a larger value .

Our homological approach mainly for the balanced case was to use the local homological condition from (2) (or(4)) (with some additional homological condition yet to be proved – this was Conjecture A) to conclude (This was Conjecture B) that is -acyclic. However, Conjecture B turned out to be false. Our new idea is to replace the local conditions on homology obtained from (2), (4) by those given by (6) which are apparently quite stronger.

We also tried to explore how to prove the main conjecture via an even stronger statement for “combinatorial cycles”. This works for the balanced case in the Erdos-Ko-Rado-regime (leading to some interesting consequences). But we did not manage to extend it beyond this regime. Domotor shoot down some half-baked attempts towards such a goal.

What we propose now is to use the following theorem instead of Conjecture A and to greatly modify Conjecture B accordingly. (For general families; being balanced is not used.)

**Theorem:** Let is a family of -sets without a sunflower of size . Then

(*) For every family of -sets which is the link of a set of size (including the case , Every set in intersect .

**Conjecture B** (greatly modified): For a family of -sets satisfying (*) we have

(**) .

This is the new plan! Below are a couple of comments.

We can, I think, translate the condition on reverse lexicographic shifting also to some conditions on homology, given in this comment, but the connection is still a little dubious and need some further checking and explanation. (Specifically, it relies on a comment I make in my paper on algebraic shifting that if is a family of -subsets of and is its complement, then the shifting of is related to the reverse lexicographic shifting of as follows: takes the complement of and apply the involution sending element to .)

If is the simplicial complex spanned by our family and is the simplicial complex spanned by the complement of we want to replace conditions of the form

(*)

by the stronger condition

(**)

For example, for a graph with vertices and edges gives all edges containing ‘1’ (which is equivalent to (*) for ) if and only if is a tree. But requiring it for implies (I think) that be a star and is equivalent (I think) to , where is the complement of .

We plan to run computer experimentation to test some of these ideas on small cases.

Let me mention that there is a variant of compound matrix, weak equivalence and of algebraic shifting (and hence also of the various homology groups we considered,) when we use symmetric products instead of exterior products. The theories based on those variants are very similar, and the advantage of the notions based on symmetric powers is that they are closer to studied notions of commutative algebra. (But I think they will be harder to compute as determinants are replaced by permanents.)

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j) **Polymath11 (?)** Tim Gowers’s proposed a polymath project on Frankl’s conjecture. If it will get off the ground we will have (with polymath10) two projects running in parallel which is very nice. (In the comments Jon Awbrey gave a links for a first in a series posts also on Frankl’s conjecture, with the catchy title, Frankl my dear.)

a) NogaFest started a few days ago. It is a wondeful meeting! My lecture entitled “polymath” refers to the older meaning of the word, so appropriate to describe Noga. (I was not aware that the word has a meaning until recently). I talked, among other things, about **polymath10**. I prepared the talk a week ahead and presented our Conjectures A and B (from polymath10 last post) hoping that perhaps I could add some positive information toward them. Well, just after my presentation was ready, I realized that Conjecture B is false. Here are the slides.

Two quotes from the lecture. First about the birthday boy: the idea of the polymath was expressed by Leon Battista Alberti (1404–1472), in the statement, most suitable to Noga **“****a man (who) can do all things if he will”.**** **Second, about polymath projects (by Gowers): “a large collaboration in which no single person has to work all that hard.”

b) Here is the link to a mathoverflow question asking for polymath proposals. There are some very interesting proposals. I am quite curious to see some proposals (perhaps mainly to see what researchers regard as central major projects,) in applied mathematics, and various areas of geometry, algebra, analysis and logic.

c) A very nice polymath proposal by Dinesh Thakur was posted by Terry Tao on the polymath blog. The task was to explain some numerically observed identities involving the irreducible polynomials in the polynomial ring over the finite field of characteristic two. David Speyer managed to prove Thakur’s observed identities! Here is the draft of the paper.

d) This reminds me that some years ago David Speyer solved a question that interested me for decades and was presented here on the blog and later on MathOverflow about systems of skew lines in three dimensional vector spaces over division rings (and especially the Quaternions).

e) Related to **polymath3**, let me mention that Michael Todd proved a small but very elegant improvement of the upper bounds by Kleitman and me from 1992. (The new bound is . The first improvement, I think, in two decades!

f) I have a very nice thing to tell you about **polymath4!** Shafi Goldwasser abstract for Nogafest talked about a new notion of randomized algorithms: A randomized algorithm to achieve a certain task (for example to find a perfect matching in a graph,) which is guaranteed to reach the same answer with high probability! Such an algorithm is called pseudo-deterministic. It is both an amazing concept, and it is quite amazing that it was not introduced before. The polymath4 question was to find deterministically a prime with n digits and A new challenge (that Shafi asks about) is to find a pseudo-deterministic efficient algorithm. Namely, a randomized algorithm which will find an n-digit prime, but with high probability the same one! (I would guess that it is still hopeless.)

g) And Terry Tao gave a beautiful lecture on Erdos discrepancy problem (the topic of **polymath5**). I understood a little better the argument (which is similar to Roth density increase argument for 3 term AP,) that allows Tao to use the logarithmically-averaged Chowla inequalities.

h) The old conjecture that centrally symmetric convex sets have nonnegative correlation w.r.t. the Gaussian distribution was proved! Let me refer you to the paper Royen’s proof of the Gaussian correlation inequality for a simple exposition of a proof by Thomas Royen, and more information on the solutions and solvers.

i) The Nogafest participants are invited to a Jazz night at Gilly’s

The third Polymath10 post is active. I hope to post a new polymath10 post in about 1-2 weeks. I hope also to return to various amazing things I am hearing on Nogafest and other places (and also on my ownfest some months ago).

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Let [] be the maximum size of a family of -sets [balanced family] with no sunflower of size . Let and .

Phillip gave a beautiful construction showing that if is finite than . This is very interesting and I found it surprising. (Earlier we only knew that .) Of course, if you can find another construction which starts with balanced families (even only those constructed by Philip) and end with exponentially larger general families (you can even enlarge k a little,) this will provide a counterexample to the Erdos-Rado conjecture.

In the previous post and some comments (like this comment and the following ones; and this comment) I continued to meditate about my homological approach. For we want to show that for balanced families, a (2m,m)-cycle will contain a sunflower with head of size smaller than m. (This will now give .) Juggeling between the homological notion of (b,c)-cycles and a combinatorial one can be useful.

Tim observed that if all pairwise intersections between sets in a sunflower free family have the same size then this leads to exponential upper bounds. This can be a starting point for an argument where “same size” is replaced by a weaker condition, or to ideas about how to construct a counterexample.

We also want to understand the expected number of sets in the sunflower-free process when we consider k-subsets fron [n]. (We would also like to understand the expected size of the union of sets in the resulting family.) This is interesting both for the general case and the balanced case. Simulations by Philip (e.g. here) and by Gonzalo (e.g. here) were presented. (I am still confused about the outcomes of the simulations, maybe we shoud run more simulations.)

In the later comments to the previous thread (starting here) that I did not digest yet, Philip offered some ideas on new constructions of various types.

Avi Wigderson asked: Is it enough to prove the Erdos-Rado conjecture for ? ? etc? Dömötör asked the following Kneser-type coloring question:

How many colors to we need to color all k-tuples of an n element set avoiding monochromatic 3-sunflowers?

Shachar mentioned in comments to post I (starting here) an exciting special case (related to matrix multiplication.)

It can be a good time to look at the classic papers by Abott, Hanson, and Sauer and , Spencer and Kostochka (for the general case; for sunflower of size three). Here is again the link for Kostochka’s survey. There are many other papers (even recent) about sunflowers and many aspect of the problem and related problems that we did not talk about.

Here are the six pages of Joel’s paper.

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Of course, there are various other avenues that can be explored: In a series of comments (e.g. this thread and that thread and this) Tim Gowers proposed a line of attack related to understanding quasirandom behavior of families of sets in terms of their pairwise intersections. (**Update**: Tim developed his ideas in further comments. A theme which is common to his approach as well as to the homological approach is to see if we can “improve” certain properties of the family after moving to an exponentially smaller subfamily. **Second update**: this post was written after 70 or so comments for post 1. There are many further interesting comments. ) Karim Adiprasito described a different topological combinatorics approach. Another clear direction is to try to extend the ideas of Spencer and Kostochka which led to the best known bounds today. Raising more ideas for attacking the conjecture is most welcome. For example, in Erdos Ko Rado theory, besides direct combinatorial arguments (mainly those based on “shifting,”) spectral methods are also quite important. Of course, the sunflower conjecture may be false as well and ideas on how to construct large families without sunflowers are also most welcome.

Terry Tao kindly set a Wiki page for the project and proposed to conduct computer experimentation for small values of and . Of course, computer experimentation will be most welcome! Some of the suggestions described below can also be tested experimentally for small values.

Let me also mention some surprising connections between the sunflower conjecture and various issues arising in matrix multiplication. As pointed by Shachar Lovett (in this comment and the following one), a counterexample to a certain structural special case of the sunflower conjecture will imply an almost quadratic algorithm for matrix multiplication!

Dömötör and I hyperoptimistically conjectured that the Erdos-Rado example is optimal for Balanced families. But Hao gave a very simple counterexample.

The status of our project at this stage is described very nicely by Tim Gowers who wrote:

At the time of writing, Gil’s post has attracted 60 comments, but it is still at what one might call a warming-up stage, so if you are interested in the problem and understand what I have written above, it should still be easy to catch up with the discussion. I strongly recommend contributing — even small remarks can be very helpful for other people, sparking off ideas that they might not have had otherwise. And there’s nothing quite like thinking about a problem, writing regular bulletins of the little ideas you’ve had, and getting feedback on them from other Polymath participants. This problem has the same kind of notoriously hard feel about it that the Erdős discrepancy problem had — it would be wonderful if a Polymath collaboration could contribute to its finally getting solved.

**Update to an earlier post.** Karim Adiprasito, June Huh, and Eric Katz have now posted their paper “Hodge theory for combinatorial geometries” which contains, among other things, a proof of the Heron-Rota-Welsh conjecture on matroids.

**Here is a reminder of the sunflower theorem and conjecture:**

A sunflower (a.k.a. Delta-system) of size is a family of sets such that every element that belongs to more than one of the sets belongs to all of them. We call the common element to all the sets the head of the sunflower (or the kernel of the sunflower), and the elements that belong to just one among the sets, the petals.

A basic and simple result of Erdos and Rado asserts that

**Erdos-Rado sunflower theorem:** There is a function so that every family of -sets with more than members contains a sunflower of size .

We denote by the smallest integer that suffices for the assertion of the theorem to be true.

**The Erdos-Rado sunflower conjecture: ** .

Here, is a constant depending on . It may be also the case that we can take for some absolute constant *C*. The conjecture is already most interesting for . I recommend to reading Kostochka’s survey paper and also, as we go, it will be useful to learn Spencer’s argument and Kostochka’s argument which made remarkable improvements over earlier upper bounds.

The main purpose of this post is to provide

**A)** **The question as an Erdos-Ko-Rado type question**

Let be the maximum size of a family of -subsets of* [n]=* containing no sunflower of size with head of size at most . (Note: it should me .)

**Basic Question**: Understand the function *f(k,r,m;n)*. Is it true that , where is a constant depending on *r*, perhaps even linear in *r*.

A family of -sets satisfies property P(k,r,m) if it contains no sunflower of size with head of size at most .

**B)** **Stars and links:** Given a family of sets and a set , the* star* of is the subfamily of those sets in containing , and the* link* of is obtained from the star of by deleting the elements of from every set in the star. Another way to say that has property P(k,r,m) is that the link of every set of size ~~at most~~ less than contains no pairwise disjoint sets.

**C)** **The balanced case**

A family of -sets is **balanced** (or -colored, or multipartite) if it is possible to divide the ground set into parts so that every set in the family contains one element from every part.

Let be the maximum size of a balanced family of -subsets of* [n]=* containing no sunflower of size with head of size at most . By randomly dividing the ground set into colors we obtain that .

**D) What we aim for.** Below we describe two variations of a homological attack on the sunflower conjecture. If successful (as they are) they will lead to the following bounds.

The first variation based on conjectured homological properties of balanced families would yield

(*)

The alternative version would give

(**)

**E) Simplicial complexes and homology**

Staring with a family we will consider the collection of sets obtained by adding all subsets of sets in . This is a simplicial complex, , and we can regard it as a geometric object if we replace every set of size by a simplex of dimension . (We call sets in of cardinality by the name** i-faces**.

The definition of homology groups only depends on the combinatorial data. For simplicity we assume that all sets in (and hence in the associated simplicial complex) are subsets of *{1,2,…,n}*. We choose a field **A **(we can agree that the field will be the field of real numbers). Next we define for i>0 the vector space of -dimensional chains as a vector space generated by i-faces of K. We also define a boundary map for every . The kernel of is the space of i-cycles denoted by ; the image of is the space of i-boundaries, denoted by . The crucial property is that applying boundary twice gives you zero, and this allows to define homology groups . The betti numbers are defined as . We will give the definition of the boundary operator further below.

**F) Acyclic families and intersecting families**

A family of -sets is acyclic if it contains no -cycle, or equivalently if . (For coefficients in , a -cycle is a family of -sets such that every set of size is included in an even number of -sets in .)

**Proposition**: An acyclic family of* k*-subsets of* [n]*, contains at most sets.

In the first post we asked: Are there some connections between the property “intersecting” and the property “acyclic?”

Unfortunately, but not surprisingly intersecting families are not always acyclic, and acyclic families are not always intersecting. (The condition from EKR theorem also disappeared.) As we mentioned in the previous post intersecting balanced families are acyclic! And as we will see for balanced families Erdos-Ko-Rado properties translate nicely into homological property.

**G) Pushing the analogy further**

We made an analogy between “intersecting” and “acyclic”. Building on this analogy

1) What could be the “homological” property analogous to “every two sets have at least *m *elements in common”?

2) What could be the “homological” property analogous to “not having *r* pairwise disjoints sets”?

I will propose answers below the fold. What is your answer?

**H) Weighted homology**

For a simplicial complex on the vertex set* [n]* the boundary operator is defined by . Where .

Given a vector of nonzero weights we can define a weighted boundary operator by

,

where . It is a simple matter of scaling a matrix that still the boundary of the boundary is zero and (over any field) the homology with respect to this weighted boundary operator does not depend on .

**I) Iterated homology**

Being acyclic guarantees that

1) What is the global homological property that will give us ?

2) What is the global homological property that will give us ?

**Answer for 1:** There is no chain in which vanishes when you apply *m* (generic) weighted boundary operators successively.

**Answer for 2:** There is no chain in which vanishes when you apply each one out of (generic) weighted boundary operators.

When both answers coincide with the top dimensional homology . For larger value of those are kind of homology-like spaces whch are called “iterated homology.”

**J) m-fold acyclic families (first try)**

Iterated homology gives us global properties of a family of sets that we want to relate to Erdos-Ko-Rado-like properties P(m,r). But in order to make such a connection we first need to study the connection between local and global properties. Here, by “local” I refer to properties described in terms of links. Let’s go back to ordinary homology and try to understand the situation when we impose (top-dimension-) acyclicity on the family as well as on links. We will start with the simplest case: what about families which are acyclic, and all links of vertices are acyclic? Let us choose the case *k=3, m=2.*

Is the number of triangles in such a 2-fold acyclic family at most linear in ? perhaps even at most ?

Here is an example with more than triangles.

But things can get much worse: Consider a Steiner triple system: namely a collection of triangles where every pair of elements is included in one triangle. It is obviously 2-acyclic and the link of every vertex is a matching and thus, it is an acyclic graph. Still, we have quadratic number of triangles.

**K) m-fold acyclic families revisited.**

**Theorem 1:** Let be a family of -sets and let be the associated -dimensional simplicial complex. Suppose that

a)

b)

and

c)

Then .

So we need a new crucial assumption: it is not enough to require that the top homology for the family and its links vanish, we need also that the th homology will vanish.

**[**One thing to keep in mind: Condition c) is not preserved when we delete sets from the family. We can hope that we can replace this condition by a weaker condition which is a monotone property relating th homology of with th homology of links of vertices. For every vertex there is a map from to . Perhaps the property is that this map is surjective for every vertex. update (Dec 13) I am less certain about what the property should be.**]**

This theorem extends also to every value of *m*.

**Theorem 2:** Let be a family of -sets and let be the associated -dimensional simplicial complex. Suppose that for every link of (including itself) whenever , then .

**L)** **Collapsibility.** An easier version of Theorem 2 is for the case that is “*d*-collapsible,” for This is a combinatorial property which is stronger than the homology condition. Using a combinatorial strong form (like collapsibility) of the homological conditions may be relevant to our case as well.

**M) A working conjecture that may assist an inductive argument.**

The following working conjecture may be useful for some inductive arguments:

**Working Conjecture: **Suppose that is a family (or just a balanced family) of -subsets of *[n]. *Suppose that for every element the star of contains a sunflower of size with head of size . Then contains a sunflower of size with head of size smaller than .

**Update:** False as stated for general families: Fano plane fails it, as Dömötör pointed out. I dont have a cunterexample for balanced families.

**Update:** False also for balanced families, as Dömötör pointed out. I dont have a counterexample (general, or better balanced) for the case . (This case might be useful.)

**N) Acyclicity and Erdos-Ko-Rado properties for balanced families.**

Now we consider the various Erdos-Ko-Rado questions for balanced families and revisit the connection to homology. For example, note that for balanced families, if then one color class has just one element hence all sets in the family contains this element.

**Proposition:** For a balanced family of -sets if every set of size is included it at least sets of size , then contains pairwise disjoint sets.

**Corollary:** If is balanced and intersecting then it is acyclic. If is balanced and has a -cycle that vanished by applying each one of generic boundary operations, then contains pairwise disjoint -sets. [corrected]

**O) The expected global consequences for balanced families without Delta systems**

**Conjecture 1 (special case of r=2):** If is a balanced family of -sets from

**Conjecture 1 (general case):** If is a balanced family of -sets from* [n]* without a sunflower of size and head with less than elements. (In other words, it has no pairwise disjoint sets for every link of a set of size less than .) Then there is no cycle in which vanishes when you apply any combination of boundary operators successively out of different (generic) boundary operators.

This last conjecture would give

(*)

**P) Avoiding coloring**

We had some difficulty to relating intersecting and acyclic families. One (conjectural) proposal was to move to balanced families. But another proposal is to relax the notion of acyclicity. (Essentially by adding additional boundary operators.)

**Theorem 4: ** (1) Let be an intersecting family of -subsets from* [n]*. Then there is no cycle in which vanishes when you apply each one out of (generic) boundary operators

(2) Let be a family of -subsets from* [n]* without pairwise disjoint sets, Then there is no cycle in which vanishes when one applies every boundary operator out of (generic) boundary operators.

(3) If F is a family of -sets from *[n]* and every two members of share at least elements then there is no cycle in which vanishes when you apply any combination of *m* boundary operators successively out of different (generic) boundary operators.

These follow directly from the fact that algebraic shifting preserves the property that is intersecting, the property that has no pairwise disjoint sets, and the property that every pairwise intersection has at least m elements.

Unfortunately, as we mentioned, the property of interest to us is not preserved under shifting. We can hope that the effect of the extra boundary operators, in an inductive argument will be as follows:

**Q) The Conjecture for the alternative direction: **

**Conjecture 2:** If is a family of -sets from* [n]* and if it has no pairwise disjoint sets for every link of a set of size less than then there is no cycle in which vanishes when you apply any combination of boundary operators successively out of different (generic) boundary operators.

This would give

(**)

**R) Shifting**

We mentioned the shifting method in this post. A collection F of* k*-subsets from* [n]={1,2,…,n}* is shifted (or compressed) if whenever a set *S* is in the collection and *R* is obtained by *S* by lowering the value of an element, then *R* is also in the family.

A shifting process is a method to move from an arbitrary family to a shifted one with the same size. Erdos, Ko and Rado described a combinatorial method for shifting in their paper on the Erdos-Ko-Rado theorem. A very basic facts from Erdos-Ko-Rado theory is

**(EKR)** P(2,m) and P(r,1) are preserved under shifting.

But not having a sunflower is not preserved under shifting. It is still possible that not having a sunflower for the family is translated to a different statement for the family obtained from it by shifting and indeed we will formulate Conjectures 1 and 2 in these terms.

**S) Algebraic shifting**

Algebraic shifting was mentioned in this post. A good reference for it is this 2002 paper.

Here is a quick definition of algebraic shifting:

(1) Start with an by generic matrix .

(2) Next consider the th-compound matrix whose entries correspond to the determinants all by minors of . Order the rows and columns of lexicographically.

(3) Given a family of subsets of* [n] *consider the submatrix of whose columns are indexed by sets in .

(4) Now, choose a basis to the rows of greedily, namely, go over the rows of one by one and add a row to the basis if it does not depend on the earlier rows.

(5) The algebraic shifting of is the family of indices for the rows in this basis.

**Theorem:** Property **(EKR)** continues to hold for algebraic shifting.

**T) Algebraic shifting and homology**

Algebraic shifting also preserves the Betti numbers as well as the dimension of various iterated homology groups.

For example, is acyclic, namely there is no -chain that vanishes when a boundary operation is applied, if and only if all sets in contains ‘1’.

There is no chain in which vanishes when you apply successively *m* weighted boundary operators if and only if all sets in contains .

There is no chain in which vanishes when you apply each one out of (generic) weighted boundary operators, if and only every set in contains an element .

**U) Our conjectures in terms of algebraic shifting**

Conjecture 1 and 2 are equivalent to:

**Conjecture 1′:** Algebraic shifting of balanced families with property P(r,m) leads to a shifted family so that every set has at least *m* elements in the set* {1,2,…,(r-1)k}.*

**Conjecture 2′:** If is a family of k-subsets of *{1,2,…,n} *with property P(r,m) then for the algebraic shifting of , every set has at least *m* elements in the set* {1,2,…,m+(r-1)k}.*

**V) Balanced shifting**

Even when dealing with balanced families we considered a shifting operation that does not preserve the balance structure.Variants for algebraic shifting for the balanced case were defined and may be useful. (EKR) is not known for balanced shifting.

**Question:** Does balanced shifting have property** (EKR)**.

**W) Methods from commutative algebra**

Methods from commutative algebra are quite powerful for demonstrating (often in another language) results about algebraic shifting and iterated homology groups.

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A sunflower (a.k.a. Delta-system) of size is a family of sets such that every element that belongs to more than one of the sets belongs to all of them. A basic and simple result of Erdos and Rado asserts that

**Erdos-Rado Delta-syatem theorem:** There is a function so that every family of -sets with more than members contains a sunflower of size .

(We denote by the smallest integer that suffices for the assertion of the theorem to be true.) The simple proof giving can be found here.

One of the most famous open problems in extremal combinatorics is:

**The Erdos-Rado conjecture: **Prove that .

Here, is a constant depending on . It may be also the case that we can take for some absolute constant *C*. The conjecture is already most interesting for . And getting progress for will already be great.

At least for the delta system conjecture is true. Every two sets form a Delta-system of size two. So . Can we find more difficult proofs and for weaker statements? The case will play a role in the context of more general questions.

An excellent review paper is Extremal problem on Δ-systems by Alexandr Kostochka. After an early paper by L. Abbott, D. Hanson, and N. Sauer imroving both the upper and lower bounds, Joel Spencer proved an upper bound of for every fixed . Spencer also proved an upper bound for . (The exponent was improved further to 1/2 by Furedi and Kahn.) A remarkable result by Sasha Kostochka from 1996 is the best upper bound known today.

**Sasha Kostochka**

Given a family of sets and a set , the* star* of is the subfamily of those sets in containing , and the* link* of is obtained from the star of by deleting the elements of from every set in the star. (We use the terms link and star because we do want to consider eventually hypergraphs as geometric/topological objects.)

We can restate the delta system problem as follows*: f(k,r)* is the maximum size of a family of *k*-sets such that the link of every set* A* does not contain *r* pairwise disjoint sets.

What can we say about families of *k*-sets from {1,2,…,n} such that that the link of every set *A* of size at most* m-1* does not contain* r* pairwise disjoint sets? In particular, what is *f(k,r,m;n)* the maximum number of sets in such a family. We can ask

**Question 1**: Understand the function *f(k,r,m;n)*.

**Question 2:** Is it true that , where is a constant depending on *r*, perhaps even linear in *r*.

My proposal is to approach the Delta-system problem via these questions. We note that Question 1 includes the Erdos-Ko-Rado theorem:

**(r=2, m=1) Erdös-Ko-Rado Theorem**: An intersecting of k-subsets of , when contains at most sets.

Here, a family of sets is “intersecting” if every two sets in the family has non empty intersection. The situation for and general was raised by Erdos-Ko-Rado (who proposed a conjecture for a certain special case), Frankl proposed a general conjecture that was settled by Alswede-Khachatrian. This was a remarkable breakthrough. The case of general r and m=1 is again a famous question of Erdos. The conjecture is that when , This is a classic result by Erdos and Gallai (1959) for graphs (k=2), and very recently it was proved for r=3 for large values of , in the paper On Erdos’ extremal problem on matchings in hypergraphs by Tomasz Luczak, and Katarzyna Mieczkowska, and for all values of by Peter Frankl.

We want much weaker results (suggested by Problem 2) than those given (or conjectured) by Erdos-Ko-Rado theory, but strong enough to apply to the Delta system conjecture.

A family of -sets is balanced (or -colored) if it is possible to color the elements with colors so that every set in the family is colorful.

**Reduction (folklore):** It is enough to prove Erdos-Rado Delta-system conjecture for the balanced case.

**Proof:** Divide the elements into d color classes at random and take only colorful sets. The expected size of the surviving colorful sets is .

A family of -sets is acyclic (with Z2 coefficient) if it contains no -cycle. A -cycle is a family of -sets such that every set of size is included in an even number of -sets in .

**Theorem 1**: An acyclic family of* k*-subsets of* [n]*, contains at most sets.

I suppose many people ask themselves:

**Question 3:** Are there some connections between the property “intersecting” and the property “acyclic?”

Unfortunately, but not surprisingly intersecting families are not always acyclic. And acyclic families are not always intersecting.(The condition from EKR theorem also disappeared in the result about acyclic families.)

Lets ignore for a minute that being acyclic and being intersecting are not related and ask.

**Question 4:** Is there some “acyclicity” condition related to (or analogous to) the property of not having 3 pairwise disjoint sets? *r* pairwise disjoint sets?

We know a few things about it.

**Question 5: **What can we say about families which are acyclic and so are all links for every set *A* of size at most* m-1?*

Here under some additional conditions there are quite a lot we can say in a direction of bounds asked for in Question 2.

(We note that one place where a connection between homology and Erdos-Ko-Rado theory was explored successfully is in the paper Homological approaches to two problems on finite sets by Rita Csákány and Jeff Kahn.)

Relating acyclicity and being intersecting is not easy in spite of the similar upper bound. We can ask now if for** ** balanced families, there are some connections between the property “intersecting” and the property “acyclic?”

**Question 6:** Let be a balanced intersecting family of -sets, is acyclic?

The answer is **yes**. Eran Nevo pointed out a simple inductive argument which also extends in various directions. If you have a balanced k-dimensional cycle (mod Z/Z2) then by induction the link of a vertex *v* (which is also a cycle) has two disjoint sets *R* and *R*‘ and taking one of those sets with *v *and the other set with yet another vertex *w* yield a disjoint pair. (Each set of size *k-1* in a cycle must be included in more than one sets of size *k; *in fact this is the only fact we are using.)

On technical matters: The project will run over this blog and Karim Adiprasito will join me in organizing it. (Perhaps to make the mathematical formulas appearing better we will move to another appearance.)

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I am happy to report on two beautiful results on convex polytopes. One disproves an old conjecture of mine and one proves an old conjecture of mine.

Lipton and Tarjan proved that a planar graph (hence the graph of a 3-polytope) with ** n** vertices can be separated to connected parts each with no more than

Lauri Loiskekoski, Günter M. Ziegler

Abstract:We show that by cutting off the vertices and then the edges of neighborly cubical polytopes, one obtains simple 4-dimensional polytopes with n vertices such that all separators of the graph have size at least This disproves a conjecture by Kalai from 1991/2004.

A major remaining open question is if we can have examples of graphs of simple 4-polytopes where every separator has at least *cn* vertices or perhaps even examples of such graphs that are expanders. (I conjecture that the answer is negative).

Expansion is closely related to diameter and there are various interesting higher dimensional analogs for the expansion property, for diameter question, and for the non-revisiting property which is crucial in the study of diameter of polytopal graphs.

Consider a sequence of polytopes that converge to a smooth convex body ** K**. It is easy that the number of vertices of these polytopes must tend to infinity and there are interesting theorems relating the quality of the approximation and the the number of vertices.

For a *d*-dimensional simplicial polytope *P* there are is remarkable set of parameters introduced by McMullen where *m=[d/2]*. is the number of vertices minus* (d+1)*. is the dimension of the space of stresses of a framework based on *P*. The nonnegativity of these numbers is the generalized lower bound theorem which is part of Stanley’s 1980 “*g*-theorem” (necessity part), which is one of the most important results on convex polytopes in the last decades. Here on the blog we devoted several posts (I,II,III,IV) to the *g*-theorem and the related *g*-conjecture for simplicial spheres. (Probably the *g*-conjecture for spheres is the single problem I devoted the most effort to in my career.)

I conjectured that for a sequence of polytopes that converge to a smooth convex body ** K, ** tends to infinity for

Karim A. Adiprasito, Eran Nevo, José Alejandro Samper

Abstract: We resolve a conjecture of Kalai relating approximation theory of convex bodies by simplicial polytopes to the face numbers and primitive Betti numbers of these polytopes and their toric varieties. The proof uses higher notions of chordality. Further, for -convex bodies, asymptotically tight lower bounds on theg-numbers of the approximating polytopes are given, in terms of their Hausdorff distance from the convex body.

The paper continues a project which led already to:

Higher chordality I. From graphs to complexes by Karim A. Adiprasito, Eran Nevo, Jose A. Samper and

Further extensions for general polytopes, and similar results regarding the nonlinear part of the *g*-theorem would be of great interest.

A **polymath10** project devoted to the **Erdos Rado Delta System Conjecture** will start over this blog in about a week. (This is one of the project proposals by Tim Gowers in this post.) Quanta magazine has an article on important progress by Maria Chudnovsky, Irene Lo, Frédéric Maffray, Nicolas Trotignon and Kristina Vušković towards an efficient coloring argorithm for perfect graphs! (BTW, while we have several notions of chordality in high dimensions I am curious about appropriate notions of perfectness.) In this MathOverflow answer, I report (and describe the background) on the 2013 paper by Zdeněk Dvořák, Jean-Sébastien Sereni, Jan Volec “Subcubic triangle-free graphs have fractional chromatic number at most 14/5“. This recent mathoverflow question asks for proposals for polymath projects.

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In 1932, Erdős conjectured:

**Erdős Discrepancy Conjecture (EDC) **[Problem 9 here] For any constant , there is an such that the following holds. For any function , there exists an and a such that

For any , the set is an arithmetic progression containing ; we call such a set, a *Homogenous Arithmetic Progression* (HAP). The conjecture above says that for any red blue coloring of the* [n] (={1,2,…,n})*, there is some HAP which has a lot more of red than blue (or vice versa). Given C*,* we let *n(C)* to be the minimum value of *n* for which the assertion of EDC holds, and given *n* we write *D(n)* as the minimum value of *C* for which* n(C) ≤ n.*

EDC was a well-known conjecture and it was the subject of the fifth polymath project (making it even more well-known,) that took place in the first half of 2010. (With a few additional threads in August-September 2012.) That *D(n) > 3* for* n > 1160* (and that *D(1160)=3* ) was proved in a 2014 paper by Boris Konev and Alexei Lisitsa.

The two defining moments in the life of a mathematical problem is the time it is born and the time it is solved. As you must have heard by now the Erdős discrepancy conjecture has recently (mid September, 2015) been proved by Terry Tao. I was very happy with the news, congratulations Terry!!

**(Video:)** Terence Tao, The Erdős Discrepancy Problem, UCLA Math Colloquium, video by IPAM, Oct 8, 2015. (Thanks to Igor Pak)

**(Papers:)** The proof of the conjecture was done in two recent papers. The first The logarithmically averaged Chowla and Elliott conjectures for two-point correlations, proves a necessary analytic number theory result related to a classical conjecture of Chowla. The second paper – The Erdős discrepancy problem, shows the derivation of EDC from the number theory result. The number theoretic paper is based on a new recent breakthrough technique in analytic number theory initiated by Matomaki and Radziwill and further studied by Matomaki, Radzwill, and Tao. I also recommend a very interesting paper Erdős and arithmetic progressions from about a year ago by Gowers: where Tim tells side-by-side the story of Erdős-Turàn problem (leading to Roth-Szemeredi’s theorem), and that of EDP.

**(Blog posts:)** The proof is described in this blog post by Tao. A similar (somewhat simpler) argument proving EDC based on a number-theoretic conjecture (The Eliott Conjecture) can be found in this very readable blog post by Tao. In 2010 Tim Gowers ran a polymath project devoted to the Erdős discrepancy problem (EDP). A concluding post following Tao’s proof on Gowers’s blog is EDP28. You can also read about the problem and its solution on Lipton and Regan’s blog and in various other places.

**(Popular scientific writings:) **Quanta magazine (Erica Klarreich) : Nature (Chris Cesare), and various other places.

Erdős’s 1957 paper on open problems in number theory, geometry, and analysis is especially interesting. (It is linked above but the link here might be more stable.) It has 15 problems in number theory, 5 problems in Geometry, and 9 problems in analysis. Some of the problems are famous, and quite a few of them were settled, but some were new to me. It would be nice to go over them and see what their status is.

We will come back to Erdős’s 57 paper at the end.

To celebrate to solution here are a few things I would like to tell you (including something about the **Erdős-Szusz discrepancy problem** and its solution), as well as more things and problems related to EDP that I am curious about (the red items):

**The proof;****Other proofs? Other applications of the methods?****The value:**What is the behavior of*D(n)?*a) Does the proof give ? What will it take to get ? b) Find a multiplicative sequence of of length*n*of +1 and -1 with discrepancy . (Even better, find an infinite sequence with this property for every n.)**Sequences with diminishing correlation to Dirichlet characters.**Find a sequence orthogonal to all characters with small discrepancy. Prove a stronger version of the conjecture for such sequences.**The hereditary discrepancy of HAP’s****Variants:**Random subsets; square free integers;**Pseudointegers**Can we understand softly and under greater generality why EDC is true?**Pseudo HAP**: A toy problem proposed by several people in polymath5 is to replace the kth HAP by a random set of integers of density 1/k. The EDC and even the $\latex {\sqrt {\log n}$ prediction should still work.**Restricted gaps**a) prime powers gaps, b) powers of two and primes gaps; c) small gaps**Modular versions****What is the strongest version of a statement saying:**Functions with values 0, 1, -1 with diminishing correlations to Dirichlet characters have large discrepancy.**Erdős-Szüsz discrepancy problem**and the question about basis. (This I heard from Gadi Kozma).-
**What is the RH-strength analog of Chowla’s conjecture?**

A few words about the proof. The part of the proof I am less unfamiliar with is the derivation of EDC from the Elliot conjecture (which refers to a slight modification of a conjecture by Eliott) that appeared in a blog post a few days before the full proof. The actual proof gives EDC from a weak “logarithmic averaged” case of the Elliot conjecture which is the subject of this paper, and proves this averaged Elliot conjecture in this paper. For completely multiplicative functions the proof has two ingredients: The first (adapted from polymath5) is to show that completely multiplicative functions which have bounded discrepancy must have diminishing correlation with every Dirichlet character. The second is the new ingredients: The Elliot conjecture asserts that for completely multiplicative functions *g* which is not correlated with a Dirichlet character, the pairwise correlation between *g(x)* and *g(x+h)* are small. If this is the case for most intervals *[n]*, then some l-2 computation shows that the discrepancy cannot be bounded.

In a very wide stroke we do have here a structure vs randomness argument like in Roth/Szemeredi theorems. But the machinery needed for the “no structure” case is quite different.

Solving the problem for completely multiplicative functions is already a big deal. For the full statement there is one more step: Terry Tao found (in polymath5) a reduction from the general question to a certain strong form of the conjecture for completely multiplicative functions with complex norm-one values. The beautiful proof for the reduction relies on “multiplicative” Fourier analysis and it is striking how “little” the proof uses.

Of course, we can ask, as always, for other proofs perhaps with weaker quantitative conclusions but applying in greater generality. Is EDC, in essence, a theorem in analytic number theory, or are there other avenues toward it? And also, as usual, we can ask if the proof or its various ingredients can be applied to other problems.

*D(n)* tends to infinity but how fast? It is reasonable to think that *D(n) *behaves asymptotically like* * . (This is suggested, among other reasons, by certain probabilistic heuristics from EDP23.)

Regarding upper bounds we can ask:

a) Find examples where the discrepancy is or even just is *o(*log* n)*. (For the best known examples the discrepancy grows like log* n*).

Regarding lower bounds that Tao’s proof gives we can ask (mainly ask Terry himself):

c) Does Tao’s proof give for some *α > 0*? How close does the proof gets to ?

One crucial step in the proof is that a completely multiplicative sequence with positive correlation with a Dirichlet character must have unbounded discrepancy. (For the actual proof the polymath5 argument needed to be extended and strengthened.) The examples with *log n* discrepancy do have positive correlation with Dirichlet characters. There are two interesting questions when the correlations with Dirichlet characters is negligible.

First we can ask if having strong diminishing correlation with all Dirichlet’s characters implies stronger lower bound on the discrepancy? My quess is that the answer is largely no.

**Question:** Find examples which have quickly diminishing correlation with all Dirichlet characters, on every interval *[n]*, were the discrepancy is still or even *polylog (n)*, or even . (Another way to put it: find a low discrepancy completely multiplicative function which really really does not look like a Dirichlet’s character.)

It is remarked in Tao’s paper that when * f(n) *is completely multiplicative, has values {-1,0,1}, has diminishing correlation with all Dirichlet characters and has positive density ρ of nonzero entries then the discrepancy is going to infinity. Probably there are interesting quantitative versions of this fact left to be explored.

Let be the hereditary discrepancy of the hypergraph of HAPs restricted to {1,2,,…,n}. In the post EDP22 an observation made by Noga Alon and myself that *g(n) *is at most , and at least roughly is shown. There were reasons to think that *g(n)* is polylogarithmically, however:

**Theorem** (Nikolov and Talwar): .

You can read about it in the paper by Nikolov and Talwar and in this blog post by Talwar. (My description of EDC above is taken from Talwar’s post.)

Here are two variants of EDP where in both we restrict ourselves to a dense set of integers.

1) Consider a random subset of integers where each integer is taken with probability *p* (independently), *0 < p < 1*.

2) Consider the subset of square free integers;

It seems reasonable that for both these variants the discrepancy is unbounded and, in fact, behaves like .

More modest but possibly already difficult tasks would be

a) Prove that in these variants the discrepancy tends to infinity.

b) Find examples where it is ; Find an algorithm to get (even heuristically) a sequence of small discrepancy.

(Yet another variant of a similar nature would be to consider completely multiplicative functions for the subset of integers involving only primes from a random subset of primes with positive density.)

An algorithm leading (heuristically) to discrepancy was described in polymath5. A better one giving (heuristically) (also to the square free variant) is discussed in this Math Overflow problem.

A toy problem proposed by several people in polymath5 is to replace the *k*th HAP by a random set of integers of density *1/k*. The EDC and even the prediction should still work (and should be easier!). Maybe Tao’s line of proof can be applied to this case even without needing the analytic number theory part.

**One of the pleasant things I learned from polymath4 was a remarkable theorem by Jeff Lagarias on Beurling primes.**

Can we understand softly and under greater generality why EDC is true? One direction is to consider pseudointegers. (Sune Kristian Jakobsen proposed and developed it in polymath5.) Pseudointegers (e.g. based on pseudoprimes) are both appealing and frustrating. Beurling primes refers to the following situation. Consider a monotone sequence of real numbers which you declare as pseudoprimes. Let us assume that, just like ordinary primes, all products (with multiplicities) of these real numbers, which we call “pseudointegers” are distinct. Now, you can ask EDP for these psudoprimes and pseudointegers. Of course, people have asked other questions about them: When will they obey the prime number theorem? or the Riemann hypothesis? This seems a very tempting direction which at the same time is also frustrating since many of the methods disappear.

You can relax the conditions for pseudoprimes: for example, allow to be only infinitesimally larger than (or, in other words, allow them values in some non-Archimedean ordered polynomial ring), you can give up the unique factorization and even allow repetitions for the primes themselves, add weights of various kind, allow complex primes, etc. etc.

There is another way to think about these pseudoprimes and integers. If you map to (the kth prime) and then the pseudointegers are in bijection with the ordinary integers albeit not in an order preserving way. So you get a new ordering on the ordinary integers. The EDP depends just on this ordering.

In polymath5 there were examples where the discrepancy for completely multiplicative assignments of pseudointegers was bounded. But I could not find it these examples. It is reasonable to think that even for an arbitrary system of pseudointegers, we can find a completely multiplicative function with discrepancy growing like polylog (n). I don’t know if Tao’s reduction extends to pseudointegers, but I did not think much about it.

I am curious also about the hereditary discrepancy for HAP for pseudoprimes. I don’t know when the upper bound of Alon and myself behaves the same as the lower bound obtained by Nikolov and Talwar. (We can pray that the upper bound is always larger than the lower bound.)

It was observed in polymath5 that for HAP with prime gaps the discrepancy can be bounded. It may hold for prime-power gaps, and as Gowers asked even for gaps which are either prime or powers of two. It is also of interest to what extent, for the quantitative versions of EDC, we can require that the gaps for the HOP be themselves small.

There were various other notions of discrepancy for hypergraphs that came from polymath5. One was a modular notion of discrepancy. We let the ground set have arbitrary non-zero weights modulo *p* and ask if there is always an edge (HAP in our case) whose elements sum to 0 modulo *p*. We do not know what the situation is for EDP (see this post), and also for Roth’s theorem for general AP (see this post). Gowers introduced yet another extension of discrepancy which again can be adapted to arbitrary discrepancy questions. He asked if we start from an arbitrary finite set *K *of irrational numbers and assign to every integer an element from K, is it always the case that the partial sums of HAP are dense modulo.

Let *S* be a set in the unit interval and let α be an irrational number. Now, consider the sequence where iff [*n*α] mod 1 belongs to *S.* The question was when is it the case that all partial sums of the sequence are bounded. Erdős and Szüsz conjectured that this is the case if and only if *S* is a finite union of intervals with endpoints integral multiples of α modulo 1. This was proved by Kesten and later other illuminating proofs were also found.

This is something I am curious about but possibly it is well understood (or meaningless).

(PNT=Prime number theorem; RH=Riemann hypothesis)

Here are a few other problems from Erdős 1957 paper. (This paper is an example of a deep matter discussed in Mathoverflow here. )

**Do you know what is the not quite hopeless question at the end? Half a century later was it solved? (I don’t know) **

And a few fragments to test you, dear readers:

**What is this about?**

**And what is this about?**

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**Update: Nov 4, 2015:** Here is the final version of the paper:** Design exists (after P. Keevash).**

On June I gave a lecture on Bourbaki’s seminare devoted to Keevash’s breakthrough result on the existence of designs. ~~Here is a draft of the paper:~~** Design exists (after P. Keevash). **

Remarks, corrections and suggestions are most welcome!

I would have loved to expand a little on

1) How designs are connected to statistics

2) The algebraic part of Keevash’s proof

3) The “Rodl-style probabilistic part” (that I largely took for granted)

4) The greedy-random method in general

5) Difficulties when you move from graph decomposition to hypergraph decomposition

6) Wilson’s proofs of his theorem

7) Teirlink’s proof of his theorem

I knew at some point in rough details both Wilson’s proof (I heard 8 lectures about and around it from Wilson himself in 1978) and Teirlink’s (Eran London gave a detailed lecture at our seminar) but I largely forgot, I’d be happy to see a good source).

8) Other cool things about designs that I should mention.

9) The Kuperberg-Lovett-Peled work

(To be realistic, adding something for half these items will be nice.)

Here is the seminar page, (with videotaped lectures), and the home page of Association des collaborateurs de Nicolas Bourbaki . You can find there cool links to old expositions since 1948 which overall give a very nice and good picture of modern mathematics and its highlights. Here is the link to my slides.

In my case (but probably also for some other Bourbaki’s speakers) , it is not that I had full understanding (or close to it) of the proof and just had to decide how to present it, but my presentation largely represent what I know, and the seminaire forced me to learn. I was lucky that Peter gave a series of lectures (Video 1, Video 2, Video3, Video4 ) about it in the winter at our Midrasha, and that he decided to write a paper “counting designs” based on the lectures, and even luckier that Jeff Kahn taught some of it at class (based on Peter’s lectures and subsequent article) and later explained to me some core ingredients. Here is a link to Keevash’s full paper “The existence of design,” and an older post on his work.

Curiously the street was named only after Pierre Curie until the 60s and near the sign of the street you can still see the older sign.

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So far there are 31 formulas and quite a few were new to me. There are several areas of combinatorics that are not yet represented. As is natural, many formulas come from enumerative combinatorics. Don’t hesitate to contribute (best – on MathOverflow) more formulas!

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