Category Archives: Combinatorics

Séminaire N. Bourbaki – Designs Exist (after Peter Keevash) – the paper

Ihp

 

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, Video3Video4 ) 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.

rue-pierre-et-marie-curie-75005-paris

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.

Important formulas in Combinatorics

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

The question collects important formulas representing major progress 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!

Here are a few:combform

Updates and plans III.

nfh

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.

Updates: Combinatorics and more

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

The Heron-Rota-Welsh conjecture was solved by Adiprasito, Huh and Katz

GilKarim  JuneHuh  Katz4

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.

Thresholds and bounds on erasure codes by Kumar and Pfister and by Kudekar, Mondelli, Şaşoğlu, and Urbanke

(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.

Explicit extractors and Ramsey numbers by Chattopadhyay and Zuckerman

(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 n bits, each with min-entropy at least \log^C n for a large enough constant C. Our extractor outputs one bit and has error n^{-\Omega(1)}. The best previous extractor, by Bourgain [B2], required each source to have min-entropy .499n.

A key ingredient in our construction is an explicit construction of a monotone, almost-balanced boolean function on n bits that is resilient to coalitions of size n^{1-\delta}, for any \delta>0. In fact, our construction is stronger in that it gives an explicit extractor for a generalization of non-oblivious bit-fixing sources on n bits, where some unknown n-q bits are chosen almost polylog(n)-wise independently, and the remaining q=n^{1-\delta} bits are chosen by an adversary as an arbitrary function of the n-q bits. The best previous construction, by Viola \cite{Viola14}, achieved q=n^{1/2 - \delta}.

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 2^{(\log \log N)^{O(1)}}-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.

The Garsia-Stanley’s decomposition conjecture was refuted by Duval, Goeckner, Klivans, and Martin

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 [S_i,F_i] where F_i are facets (maximal faces).

A Whitney Trick for Tverberg-Type Problems by Mabillard and Wagner

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 R^d 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).

Behavior of Möbius functions (and other multiplicative functions) in short intervals by Matomaki and Radziwill

(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)).

Belated updates : Past and future events

My fest

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.

DRG_1400 Douglas Guthrie Photos DRG_1468 Douglas Guthrie PhotosDRG_1567 Douglas Guthrie Photos DRG_1605 Douglas Guthrie PhotosDRG_1692 Douglas Guthrie Photos DRG_1719 Douglas Guthrie PhotosDRG_1839 Douglas Guthrie Photos DRG_1724 Douglas Guthrie PhotosDRG_1871 Douglas Guthrie Photos DRG_1775 Douglas Guthrie Photos  DRG_1910 Douglas Guthrie Photos DRG_1837 Douglas Guthrie Photos  DRG_1906 Douglas Guthrie Photos DRG_1988 Douglas Guthrie Photos DRG_2049 Douglas Guthrie Photos DRG_2114 Douglas Guthrie Photos DRG_2134 Douglas Guthrie Photos DRG_2224 Douglas Guthrie Photos  DRG_1586 Douglas Guthrie Photos DRG_1809 Douglas Guthrie PhotosGil3  DRG_1863 Douglas Guthrie Photos

Travels

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

BCC2015

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.

Bourbaki seminaire: Designs after Keevash

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.

LFT100  and  celebrating (small part of) Matousek’s work

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.

More birthday conferences

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.

Guest posts by Thilo Weinert

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

Polymath plans

I plan a new polymath project. Details will follow.

Between two cities

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.

WV1  WV2

view

Checking the 2008 and 2011 plans: (In red – unfulfilled plans, in Green fulfilled plans).

2008 Plans and updates

Continue reading

NogaFest, NogaFormulas, and Amazing Cash Prizes

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 !

nogafest Cash Prizes!

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.

More cash prizes!!

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.

And even more cash prizes !!!

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.

And Even More! Win a Travel Grant to the conference

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

Rules

People involved in preparing the poster are not eligible.

The competition opens now!!!

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

Choongbum Lee proved the Burr-Erdős conjecture

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

Ramsey’s famous theorem asserts that if H is a complete graph on m vertices then R(H) is finite.   Ir follows that R(H) is finite for every graph H and understanding the dependence of R(H) on H 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 d-degenerate if it can be reduced to the empty graph by successively deleting vertices of degree at most d. 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 d, there exists a constant c = c(d) such that every d-degenerate graph H on n vertices satisfies r(H)\le cn.  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!

More Reasons for Small Influence

influencecartoon1

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 Inf(f) <= (O(\log(S)))^{d-1} 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. 

Boppana-Hastad-Linial-Mansour-Nisan-Boppana-Hastad

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 (\log s)^{d-1}.

The total influence I(f) of f is defined as follows: for an input x write I(x) for the number of neighbors y of x with f(y) \ne f(x). I(f) = \mathbb EI(x).

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).

Sipser MIT photo

Mike Sipser

Inverse Boppana-Hastad-LMN

Conjecture: (Benjanmini, Kala, and Schramm, 1999): Every Boolean function  f  is “close” to some depth-d size s circuit with (\log s)^{d-1} 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 ((log size)^{depth-1} ) 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) logn (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 (log s)^d 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.