The purpose of this short but valuable post is to bring to your attention

**Igor Pak’s **Collection of Combinatorics Videos

The purpose of this short but valuable post is to bring to your attention

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. Continue reading

**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.

Another spin-off of the Noga-poster-formula-competition is a MathOverflow question: Important formulas in combinatorics.

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!

**Update on the great Noga’s Formulas competition.** (Link to the original post, many cash prizes are still for grab!)

This is the third “Updates and plans post”. The first one was from 2008 and the second one from 2011.

A lot is happening! I plan to devote special posts to some of these developments.

**Karim Adiprasito (with a fan), June Huh, and Eric Katz (click to enlarge!)**

The Heron-Rota-Welsh conjecture regarding the log-concavity of coefficients of the characteristic polynomials of matroids is now proved in full generality by Karim Adiprasito, June Huh, and Erick Katz! (Along with several other related conjectures.) A few years ago Huh proved the conjecture for matroids over the reals, and with Katz they extended it to representable matroids over any field. Those results used tools from algebraic geometry. (See this post and this one.) Some months ago Adiprasito and Sanyal gave a proof, based on Alexanderov-Fenchel inequalities and measure concentration, for $c$-arrangements. The general approach of Adiprasito, Huh and Katz of doing “algebraic geometry” in more general combinatorial contexts is very promising. Here is a link to a vidotaped lecture Hodge theory for combinatorial geometries by June Huh.

(Thanks to Elchanan Mossel and Avi Wigderson for telling me about it.)

Reed-Muller Codes Achieve Capacity on Erasure Channels by Santhosh Kumar, Henry D. Pfister

(and thanks to Kodlu’s comment) Reed-Muller Codes Achieve Capacity on the Binary Erasure Channel under MAP Decoding, by Shrinivas Kudekar, Marco Mondelli, Eren Şaşoğlu, Rüdiger Urbanke

Abstract (for the first paper; for the second see the comment below): This paper introduces a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding. Rather than relying on the precise structure of the codes, this method requires only that the codes are highly symmetric. In particular, the technique applies to any sequence of linear codes where the blocklengths are strictly increasing, the code rates converge to a number between 0 and 1, and the permutation group of each code is doubly transitive. This also provides a rare example in information theory where symmetry alone implies near-optimal performance.

An important consequence of this result is that a sequence of Reed-Muller codes with increasing blocklength achieves capacity if its code rate converges to a number between 0 and 1. This possibility has been suggested previously in the literature but it has only been proven for cases where the limiting code rate is 0 or 1. Moreover, these results extend naturally to affine-invariant codes and, thus, to all extended primitive narrow-sense BCH codes. The primary tools used in the proof are the sharp threshold property for monotone Boolean functions and the area theorem for extrinsic information transfer functions.

For me, a pleasant surprise was to learn about connections between threshold behavior and coding theory that I was not aware of, and here specifically, using results with Bourgain on influences under specific groups of permutations.

(Thanks to Guy Kindler and Avi Wigderson.)

Explicit Two-Source Extractors and Resilient Functions, by Eshan Chattopadhyay and David Zuckerman

Abstract: We explicitly construct an extractor for two independent sources on bits, each with min-entropy at least for a large enough constant . Our extractor outputs one bit and has error . The best previous extractor, by Bourgain [B2], required each source to have min-entropy .

A key ingredient in our construction is an explicit construction of a monotone, almost-balanced boolean function on bits that is resilient to coalitions of size , for any . In fact, our construction is stronger in that it gives an explicit extractor for a generalization of non-oblivious bit-fixing sources on bits, where some unknown bits are chosen almost -wise independently, and the remaining bits are chosen by an adversary as an arbitrary function of the bits. The best previous construction, by Viola \cite{Viola14}, achieved .

Our other main contribution is a reduction showing how such a resilient function gives a two-source extractor. This relies heavily on the new non-malleable extractor of Chattopadhyay, Goyal and Li [CGL15].

Our explicit two-source extractor directly implies an explicit construction of a -Ramsey graph over $N$ vertices, improving bounds obtained by Barak et al. [BRSW12] and matching independent work by Cohen [Coh15b].

Here are comments by Oded Goldreich. For me, a pleasant surprise regarding the construction is that it uses, in addition to an ingenious combination of ingenious recent results (by Li, Cohen, Goyal, the authors, and others) about extractors, also influences of sets of Boolean functions and, in particular, the important construction of Ajtai and Linial. (that I mentioned here several times). Recently with Bourgain and Kahn we studies influences of large sets giving examples related to the Ajtai-Linial example. **Update**: Another pleasant surprise was to learn (from Avi W.) that among the ingredients used in this new work is Feige’s collective coin flipping method with a very small number of rounds, which was used by Li miraculously in the extractor engineering.

A non-partitionable Cohen-Macaulay simplicial complex by Art M. Duval, Bennet Goeckner, Caroline J. Klivans, and Jeremy L. Martin.

Duval, Goeckner, Klivans, and Martin gave an explicit and rather small counterexample to a conjecture of Garsia and Stanley that every Cohen-Macaulay simplicial complex is decomposable, namely its set of faces can be decomposed into Boolean intervals where are facets (maximal faces).

The much awaited paper by Mabillard and Wagner is now on the arxive. See this post on topological Tverberg’s theorem.

Eliminating Higher-Multiplicity Intersections, I. A Whitney Trick for Tverberg-Type Problems, by Isaac Mabillard and Uli Wagner

**Abstract:** Motivated by topological Tverberg-type problems and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into R^d without triple, quadruple, or, more generally, r-fold points. Specifically, we are interested in maps f from K to that have no r-Tverberg points, i.e., no r-fold points with preimages in r pairwise disjoint simplices of K, and we seek necessary and sufficient conditions for the existence of such maps.

We present a higher-multiplicity analogue of the completeness of the Van Kampen obstruction for embeddability in twice the dimension. Specifically, we show that under suitable restrictions on the dimensions, a well-known Deleted Product Criterion (DPC) is not only necessary but also sufficient for the existence of maps without r-Tverberg points. Our main technical tool is a higher-multiplicity version of the classical Whitney trick.

An important guiding idea for our work was that sufficiency of the DPC, together with an old result of Ozaydin on the existence of equivariant maps, might yield an approach to disproving the remaining open cases of the long-standing topological Tverberg conjecture. Unfortunately, our proof of the sufficiency of the DPC requires a “codimension 3” proviso, which is not satisfied for when K is the N-simplex.

Recently, Frick found an extremely elegant way to overcome this last “codimension 3” obstacle and to construct counterexamples to the topological Tverberg conjecture for d at least 3r+1 (r not a prime power). Here, we present a different construction that yields counterexamples for d at least 3r (r not a prime power).

(Thanks to Tami Ziegler) We followed over here here sparsely and laymanly a few developments in analytic number theory (mainly related to gaps in primes and Möbius randomness). It is a pleasure to mention another breakthrough, largely orthogonal to earlier ones by Kaisa Matomaki and Maksym Radziwill. (Here is a link to the paper and to related blog posts by Terry Tao (1), (2) (reporting also on subsequent works by Matomaki, Radzwill, and Tao) NEW (3) (4)).

**Update** (Sept 18, 2015): Terry Tao have just uploaded a paper to the arxive where he solves the Erdos Discrepancy problem! The number theory works by Tao with Matomaki and Radzwill play important role in the proof. See this blog post on Tao’s blog with links to two new relevant papers, and this post by Tim Gowers.

On mid-June my former students organized a lovely conference celebrating my 60th birthday which I enjoyed greatly. I do plan to devote a post to the lectures and the event. Meanwhile, here are a few pictures.

In the last year or so I made only very short trips. Here is a quick report on some from the last months.

This was the second time I participated in a British combinatorial conference, after BCC1979 that I participated as a student. My lecture and paper for the proceedings deal with questions around Borsuk’s problem. Here is the BCC paper Some old and new problems in combinatorial geometry I: Around Borsuk’s problem. The proceeding is as always very recommended and let me mention, in particular, Conlon, Fox and Sudakov’s survey on Graph Ramsey theory. One of the participants, Anthony Hilton, took part in each and every earlier BCC. Another, Peter Cameron (blog) also gave an impressive singing with guitar performance.

I gave an expose on Keevash’s work about designs. My experience with giving this seminar is quite similar to the experience of other mathematicians. It was an opportunity to learn quite a few new things. Here is a draft of the written exposition Design exists (after Peter Keevash). . (And here are the slides) Remarks are most welcome. The event was very exciting and J-P Serre actively participated in the first half of the day. I plan to write more about it once the paper is finalized.

Laszlo Fejer Toth 100th birthday conference was in Budapest. I gave a talk (click for the slides) on works of Jiri Matousek. It was great to meet many friends from Hungary and other places, some of which I did not meet for many years, including Asia Ivic-Weiss, Wlodek and Greg Kuperberg, Frank Morgan, Sasha Barvinok, and many others. I plan to report at a later time on some things Sasha Barvinok have told me.

My colleagues Abraham Neyman (Merale) and Sergiu Hart celebrated with a back-to-back conferences devoted to Game theory. Egon Schulte and Caroly Bezdek celebrated together a 60th birthday conference. Congratulations to all.

On infinite combinatorics are coming. We have some further promises for guest posts and even guest columns.

I plan a new polymath project. Details will follow.

We live now in Tel-Aviv and I commute 2-3 times a week to Jerusalem. Jerusalem is, of course, a most exciting and beautiful city and a great place to live (especially in the summer), and I also love Tel-Aviv, its rhythm and atmosphere, and the beach, of course. My three children and grandchild are TelAvivians. One interesting aspect of the change is the move from a ground floor with a yard to a high floor with view.

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Ladies and gentlemen, a conference celebrating Noga Alon’s 60th birthday is coming on January. It will take place at Tel Aviv University on January 17-21. Here is the event webpage. **Don’t miss the event !**

The poster includes 15 formulas representing some of Noga’s works. Can you identify them?

The first commentator to identify a formula will win a prize of 10 Israeli Shekels (ILS) that can be claimed on Noga’s Fest itself, (or else, in person, next time we meet after the meeting.) Cash prizes claimed in person on the meeting will be doubled! Cash prizes for the oldest and newest formulas are tripled! There is a limit of one answer/prize per person/ per week. Answers need to include the formula itself, tell what the formula is, and give crucial details about it.

For each of these formulas, once identified, the comment giving the latest place where the formula is reproduced, (in a later paper or book not coauthored by any of the original discoverers) will be eligible also to 5 ILS prize. The same doubling and tripling rules as above apply. Here there is no limit on answers per person.

There will be 5 additional prizes of 20 ILS for formulas by Noga, that did not make it to the poster. Same doubling and tripling rules apply.

Among all participants who are students or post docs, one grant for a round trip to the meeting will be given.

People involved in preparing the poster are not eligible.

And here are more details on the meeting itself. (The meeting also celebrates a decade anniversary for Zeilberger’s Opinion 71.) Continue reading

My beloved mother Carmella Kalai passed away last week.

With me, 1956

My father Hanoch Kalai, my mother Carmella, My sister Tamar (Tami) and me around 1957). Continue reading

Let be a graph. The Ramsey number is the smallest such that whenever you color the edges of the complete graph with vertices with two colors **blue** and **red**, you can either find a blue copy or a red copy of .

Ramsey’s famous theorem asserts that if is a complete graph on vertices then is finite. Ir follows that is finite for every graph and understanding the dependence of on is a very important question. Of course there are very basic extensions: to many colors, to different requirements for different colors, and to hypergraphs.

A graph is -degenerate if it can be reduced to the empty graph by successively deleting vertices of degree at most . Thus, trees are 1-degenerate (and 1-degenerate graphs are forests), and planar graphs are 5-degenerate. For graphs to be degenerate is equivalent to the condition that the number of edges is at most linear times the number of vertices uniformly for all subgraphs.

In 1973, Burr and Erdős conjectured that that for every natural number , there exists a constant such that every -degenerate graph on vertices satisfies This is a very different behavior than that of complete graphs where the dependence on the number of vertices is exponential. In 1983 Chvátal, Rödl, Szemerédi, and Trotter proved the conjecture when the maximum degree is bounded. Over the years further restricted cases of the conjectures were proved some weaker estimates were demonstrated. These developments were instrumental in the developments of some very basic tools in extremal and probabilistic combinatorics. Lee’s paper Ramsey numbers of degenerate graphs proved the conjecture!

Readers of the big-league ToC blogs have already heard about the breakthrough paper An average-case depth hierarchy theorem for Boolean circuits by Benjamin Rossman, Rocco Servedio, and Li-Yang Tan. Here are blog reports on Computational complexity, on the Shtetl Optimized, and of Godel Lost letter and P=NP. Let me mention one of the applications: refuting a 1999 conjecture by Benjamini, Schramm and me.

**Update:** Li-Yang Tang explained matters in an excellent comment below. Starting with: “In brief, we believe that an average-case depth hierarchy theorem rules out the possibility of a converse to Hastad-Boppana-LMN when viewed as a statement about the total influence of *constant*-depth circuits. However, while the bound is often applied in the setting where *d* is constant, it in fact holds for all values of *d*. It would interesting to explore the implications of our result in regimes where *d* is allowed to be super-constant.

Let me add that the bounded depth case is an important case (that I referred to here), that there might be some issues failing the conjecture for non-constant depth “for the wrong reasons”, and that I see good prospect that RST’s work and techniques will refute BKS conjecture in full also for non-bounded depth.

**Update: **Rossman, Servedio, and Tan refuted some important variations of our conjecture, while other variations remain open. My description was not so accurate and in hindsight I could also explained the background and motivation better. So rather than keep updating this post, I will write a new one in a few weeks.

**Theorem:** If* f* is described by a bounded depth circuit of size s and depth *d* then* I(f)* the total influence of* f*, is at most .

The total influence of is defined as follows: for an input write for the number of neighbors *y* of *x* with . .

The history of this result as I remember it is that: it is based on a crucial way on Hastad Switching lemma going back to Hastad 1986 thesis, and for monotone functions one can use an even earlier 1984 result by Boppana. It was first proved (with exponent “d”) in 1993 by Linial-Mansour and Nisan, as a consequence of their theorem on the decay of Fourier coefficients for AC0 functions, (also based on the switching lemma). With the correct exponent d-1 it is derived from the switching lemma in a short clean argument in a 97 paper by Ravi Boppana; and finally it was extended to a sharpening of LMN result about the spectral decay by Hastad (2001).

**Mike Sipser**

**Conjecture:** (Benjanmini, Kala, and Schramm, 1999): Every Boolean function *f* is “close” to some depth-*d* size *s* circuit with not much larger than* I(f).*

Of course, the exponent *(d-1)* is strongest possible but replacing it with some constant times *d* is also of interest. (Also the monotone case already capture much interest.)

As we will see the conjecture is false even if the exponent* d-1* is replaced by a constant times *d*. I do not know what is the optimal function *u(d) *if any for which you can replace the exponent *d-1* by *u(d)*.

**Update:** Following some comments by Boaz Barak I am not sure that indeed the new examples and results regarding them leads to disproof of our conjecture. The remarkable part of RST’s paper is that the RST example cannot be approximated by a circuit of smaller depth – even by one. (This has various important applications.) In order to disprove our conjecture one need to show that the influence of the example is smaller than what Boppana’s inequality ( ) gives. This is not proved in the paper (but it may be true).

The RST’s result **does say** that if the influence is (say) log*n* (where *n* is the number of variables,) and the function depends on a small number of variables then it need not be correlated with a function in AC0.

Anyway I will keep you posted.

in 2007 O’Donnell and Wimmer showed that our inverse conjecture is false as stated. They took a Boolean function which is a tribe function on half the variables and “anti-tribes” on the rest. This still left the possibility that the exponent *d-1* could be replaced by *d* or that “close” could be replaced by a weaker conclusion on substantial correlation.

Rossman, Servedio, and Tan.show a genuinely new reason for small influence!Their example, named after Mike Sipser, is based on the AND-OR tree – a Boolean formula with alternating AND and OR levels and carefully designed parameters. The crucial part is to show that you cannot approximate this function by lower depth circuits. The theorem proved by RST is amazingly strong and does not allow reducing the depth even by one! The novel technique for proving it of random projections is very exciting.

It is still possible (I think) that such inverse theorems hold when the individual influences of all variables is below *polylog(n)/n* where *n* is the number of variables. Let me pose it as a conjecture:

**Conjecture:** Every Boolean function *f* with *n* variables and individual influences below *polylog (n)/n* is close to a function *g* ~~in AC0 ~~ of size *s* depth *d* where is polylog (n).

And here is a post on TCSexchange with a question about “monotone vs positive” for the class **P**. Similar questions for AC0 and TC0 were asked in this post.

It is a pleasure to announce my own birthday conference which will take place in Jerusalem on June 15-16 2015.

Here is the meeting’s homepage!

The organizers asked me also to mention that some support for accommodation in Jerusalem for the duration of the conference is available.

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