Consider an -step simple random walk (SRW) on a Cayley graph of a finitely generated infinite group . Refresh independently each step with probability , to get from . Are there groups for which at time the positions and are asymptotically independent? That is, does the (total variation) distance between the chain and two independent copies go to 0, as ?

Note that on the line , they are uniformally correlated, and therefore also on any group with a nontrivial homomorphism to , or on any group that has a finite index subgroup with a nontrivial homomorphism to . On the free group and for any non-Liouville group, and are correlated as well, but for a different reason: both and have a nontrivial correlation with .

Itai Benjamini and Jeremie Brieussel conjecture that these are the only ways not to be noise sensitive. That is, if a Cayley graph is Liouville and the group does not have a finite index subgroup with a homomorphism to the reals, then the Cayley graph is noise sensitive for the simple random walk. In particular, the Grigorchuk group is noise sensitive for the simple random walk!

“Physically, an *ℓ*-noise sensitive process can somewhat not be observed, since the observation does not provide any significant information on the actual output . Speculatively, this could account for the rarity of Liouville groups in natural science. Indeed besides virtually nilpotent ones, all known Liouville groups are genuinely mathematical objects .”

An update on polytope integrality gap: In my ICM paper and also in this post I asked the beautiful problem that I learned from Anna Karlin if for vertex cover for every graph G and every vector of weights, there is an efficient algorithm achieving the “polytope integrality gap”. Anna Karlin kindly informed me that Mohit Singh got in touch with her after seeing the conjecture on my blog and pointed out that the hope for approximating the polytope integrality gap for vertex cover is unlikely to be possible because of its relationship to fractional chromatic number. Mohit noted that fractional chromatic number is hard to approximate even when it is constant assuming UGC. I still think that the notion of polytope integrality gap for vertex cover as well as for more general problems is important and worth further study.

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There are finitely many convex lattice polygons contained in the square. Their number turns out to be

This is a large number. How does a typical element of this large set look? Is there a

limit shape of these convex lattice polygons?

To answer this question it is convenient to consider the lattice and define as the family of all convex -lattice polygons lying in the unit square . The polygons in have a limit shape (as ) if there is a convex set such that the overwhelming majority of the polygons in are very close to . In other words, for every the number of polygons in that are farther than from (in Hausdorff distance, say) is a minute part of , that is, as . To put it differently, the average of the characteristic functions of tends to a zero-one function:

The limit shape theorem says that such a exists, its boundary consists of four parabola arcs each touching consecutive sides of at their midpoints, see the figure.

The proof is based on the fact that a convex (lattice or non-lattice) polygon with vertices (in this order on its boundary) is uniquely determined by the edge-vectors . Using this one can write down the generating function of the number of convex lattice paths from to lying in the triangle whose vertices are and . And this number can be estimated by saddle point methods (from complex variables). This is also how formula (1) for can be established.

On the geometry part one needs a beautiful (and almost elementary) result saying that, in a triangle with vertices and with subtriangles and (see the figure)

with equality iff the line segment 46 is touches the special parabola arc at point 5. The special parabola arc is the one that touches sides 12 and 13 of at points 2 and 3. I was very proud of inequality (2) but it turned out that it had been known for long (cf Blaschke: Vorlesungen Über Differentialgeometrie II, (1923) page 38).

In the proof one needs a slightly stronger version of (2). Assuming point 4 (resp. 6) divides segment 12 (and 13) in ratio , (and ), the stronger inequality says that

which was probably not known to Blaschke.

It is important to point out that is the unique convex subset of whose affine perimeter is the largest among all convex subsets of . I’ll return to the affine perimeter later.

Yakov Sinai came up with a different, elegant, and more powerful proof using canonical ensembles from statistical physics. His method was developed further by Vershik and Zeitouni, by Bureaux and Enriquez, by Bogachev and Zarbaliev.

More generally, one can consider a convex body (compact convex set with non-empty interior) in the plane and the family of all convex -lattice polygons contained in , and ask whether a similar limit shape exists in this case. The answer is yes. The limit shape, , is the unique convex subset of whose affine perimeter is maximal among all convex subsets of . The affine perimeter is upper semicontinuous, implying the existence of convex subset of with maximal affine perimeter. The proof of its uniqueness requires extra effort. In the case when is the unit square , the limit shape is equal to . Note that for every convex body with , .

What happens if, instead of the -lattice points in , we take a random sample of points from , chosen independently and uniformly? Let be the set of all polygons whose vertices belong to . This is again a finite set and one can show that

where is equal to the affine perimeter, , of . This confirms the philosophy (or my intuition) that random points and lattice points in convex bodies behave similarly. Note that is of order while is of order which is fine as number of -lattice points in is approximately .

Even more interestingly, the limit shape of the polygons in is , in the sense that, in expectation, the overwhelming majority of polygons in is very close to . More precisely, for every

where stands for the Hausdorf distance.

The proof of the lattice case does not work here. The edge vectors determine the convex polygon, still, but the edge vectors can’t be used, there is no generating function, etc. Instead the proof is based on the following two theorems. For the first let be a triangle with two specified vertices and say, and let be a random independent sample of uniform points from . Then is called a convex chain (from to ) if the convex hull of is a convex polygon with exactly vertices. Then

a surprisingly precise result (due to Pavel Valtr).

For the second theorem let denote the probability that the random sample (independent and uniform again) lands in convex position, that is, their convex hull is a convex -gon. For this is Sylvester’s famous four point problem from 1864 (although he did not specify the underlying convex body ). Then

with the same as before. The proof of this theorem uses the previous result of Valtr about convex chains in triangles and the properties of , the largest affine area convex subset of . The set appears again: it is the limit shape of under the condition that landed in convex position.

The map is affinely equivariant and has interesting properties. turns out to be the limit shape in some further cases as well. For instance, the maximal number of vertices of the polygons in equals

as . This is of course the same as the maximal number of points in that are in convex position. Although the convex -lattice polygon with maximal number of vertices is not necessary unique, they have a limit shape which is again . The same happens in as well. In this case, however, the expectation of the maximal number of vertices is equal to constant times but the value of this (positive) constant is not known. The reason is the following. In the triangle with specified vertices and , and random sample we define as the maximal number of points from that form a convex chain in from to . The random variable is concentrated around its expectation, which is equal to some non-negative constant times as but this constant is not known. Experiments suggest that it is equal to 3 but there is no proof in sight. Not surprisingly, the limit shape of these maximal convex chains is again the special parabola arc in . This question about the random variable is similar to the longest increasing subsequence problem but much less is known about it.

What remains of the limit shape phenomenon in higher dimensions? Well, hardly anything has been proved. In the simplest case, let denote the unit cube in 3-space, and let denote the set of all convex -lattice polygons contained in . It is known that is between and with , but nothing more precise. Probably there is a limit shape here as well, and it might be the convex subset of that has the largest affine surface area. The existence of such a set follows the same way as above but its uniqueness is not known.

Finally a few words about the affine perimeter. Given a convex curve in the plane, choose points on it, take the tangent lines at these points and form the triangles as in the figure. By definition, the affine perimeter is the infimum of the sum as the subdivision gets finer and finer. The affine perimeter of the unit circle is which explains the constant in front of . The exponent is the right choice here: for larger exponent the sum is zero, and for smaller it is infinity (for the circle for instance). Inequality (2) shows that infimum in the definition can be replaced by limit. The affine perimeter of a convex polygon is zero.

For a twice differentiable curve where is the curvature and the radius of curvature and means integration with arc length. The affine perimeter is an affine invariant or rather equivariant meaning that for a non-degenerate affine transformation . Quite often the affine perimeter (and the affine surface area) appears in connection with affine equivariant properties of the convex set . One example is best approximation by inscribed polygons on vertices. When approximation is measured by then the best approximating polygon satisfies the estimate

The set , the convex subset of with maximal affine perimeter has interesting properties. For instance its boundary contains no line segment, and if some piece of its boundary lies in the interior of , then this piece is a parabola arc. It has positive curvature everywhere. It is of course affinely equivariant meaning that . According to (3) has the worst approximation properties among all convex subsets of . This might explain why it comes up as the limit shape so often. Actually, the high dimensional analogue of (3) suggests that the limit shape in higher dimensions is again connected to the maximal affine surface area subset of the underlying convex body.

More reading: Imre Bárány, Random points and lattice points in convex bodies, Bull AMS (2008)

]]>Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture

Contains an amazingly short and beautiful proof of a famous open problem from the theory of computing – the sensitivity conjecture posed by Noam Nisan and Mario Szegedi

**Hao Huang**

**Abstract: **In this paper, we show that every -vertex induced subgraph of the -dimensional cube graph has maximum degree at least . This result is best possible, and improves a logarithmic lower bound shown by Chung, Füredi, Graham and Seymour in 1988. As a direct consequence, we prove that the sensitivity and degree of a boolean function are polynomially related, solving an outstanding foundational problem in theoretical computer science, the Sensitivity Conjecture of Nisan and Szegedy.

The proof relies on important relation between the two problems by Gotsman and Linial. It uses beautifully Cauchy’s interlace theorem (for eigenvalues).

Thanks to Noga Alon for telling me about the breakthrough. For more on the conjecture and a polymath project devoted to solve it see this post on Aaronson’s Shtetl Optimized (SO). See also this post on GLL. **Updates:** See this new lovely post on SO (and don’t miss the excellent comment thread); and this post by Boaz Barak on WOT (Window on Theory) explains the main ingredient of Huang’s proof. Here is a post by Lance Fortnow on CC (Computational Complexity) mentioning a remarkable tweet by Ryan O’Donnell (see below the fold). Here is a comment by Hao on SO on the history of his own work on the problem since he heard it from Mike Saks in 2012. **More**: And here is a lovely new post by Ken Regan on GLL, and here on Fedor Petrov’s new blog (congratulations, Fedya!) you can see the proof without the interlace theorem (as was also noted by Shalev Ben David); and here on CC Lance explains the proof of the Gotsman Linial reduction. (On Fedor’s blog you can find a combinatorial proof that e⩽π!)

Also today on the arXive a paper by Zuzana (Zuzka) Patáková and me: Intersection Patterns of Planar Sets. Like one of my very first papers “Intersection patterns of convex sets” the new paper deals with face numbers of nerves of geometric sets.

]]>**Annika Heckel**

Sorry for the long period of non blogging. There are a lot of things to report and various other plans for posts and I hope to come back to it soon. But it is nice to break the silence with another sensational result by Annika Heckel. I first heard about it some time ago and Noam Lifshitz just informed be that the paper is on the arXive!

And here is the **abstract:** We show that the chromatic number of $latex G(n,1/2)$ is not concentrated on fewer than consecutive values. This addresses a longstanding question raised by Erdős and several other authors.

The Introduction tells the history of the problem very nicely.

]]>Hedetniemi’s 1966 conjecture asserts that if and are two graphs, then the chromatic number of their tensor product equals the minimum of their individual chromatic numbers. Here, the vertex set of is the Cartesian product of and and two vertices and are adjacent if is adjacent to and is adjacent to . (mistake corrected.) Every coloring of induces a coloring of , and so is every coloring of . Therefore, . Hedetniemi conjectured that equality always hold and this is now refuted by by Yaroslav Shitov.

The example and the entire proof are quite short (the entire paper is less than 3 pages; It is a bit densely-written).

** Yaroslav Shitov **

To tell you what the construction is, I need two important definitions. The first is the notion of the exponential graph .

The exponential graph arose in the study of Hedetniemi’s conjecture in a 1985 paper by El-Zahar and Sauer. The vertices of are all maps from to . Two maps are adjacent if whenever are adjacent vertices of then . El-Zahar and Sauer showed that importance of the case that is a graph and is an exponential graph of for Hedetniemi’s conjecture. (The entire conjecture reduces to this case.) It is thus crucial to study coloring of exponential graphs which is the subject of the three claims of Section 1 of Shitov’s paper.

The second definition is another important notion of product of graphs: The strong product *G* ⊠ *H *of two graphs and . The set of vertices is again the Cartesian product of the two sets of vertices. This time, and are adjacent in *G* ⊠ *H* if either

(a) is adjacent to and is adjacent to

OR

(b) is adjacent to and or and is adjacent to

(The edges of condition (a) are the edges of the *tensor product* of the two graphs and the edges of condition (b) are the edges of the *Cartesian product* of the two graphs.)

For Shitov’s counterexample given in Section 2 of his paper, is the strong product of a graph with girth at least 10 and fractional chromatic number at least 4.1 with a large clique of size . The second graph is the exponential graph . Put . Shitov shows that when is sufficiently large then the chromatic number of both is , but the chromatic number of their tensor product is smaller than .

(Have a look also at Yaroslav’s other arXived papers! )

Let me make one more remark. (See the Wikipedea article.) The infinite version of Hedetniemi’s conjecture was known to be false. Hajnal (1985) gave an example of two infinite graphs, each requiring an uncountable number of colors, such that their product can be colored with only countably many colors. Rinot (2013) proved that in the constructible universe, for every infinite cardinal , there exist a pair of graphs of chromatic number greater than , such that their product can still be colored with only countably many colors. (Here is the paper.) Is there a relation between the finite case and the infinite case? (Both theories are quite exciting but direct connections are rare. A rare statement where the same proof applies for the finite and infinite case is the inequality .

Here is a link to a survey article by Claude Tardif, (2008), “Hedetniemi’s conjecture, 40 years later” .

A few more thing worth knowing:

1) The weak version of the conjecture that asserts that If , then where tends to infinity with is still open.

2) Xuding Zhu proved in 2011 that the fractional version of the conjecture is correct,

3) The directed version of the conjecture was known to be false (Poljak and Rodl, 1981).

4) The conjecture is part of a rich and beautiful theory of graph homomorphisms (and the category of graphs) that I hope to come back to in another post.

]]>Here is what you voted for

**TYI37 poll: Final-results**

Analysis of the poll results: Almost surely 2 is the winner with 30.14% of the 209 votes, and almost surely infinity (28.71%) comes close at second place. In the third place is almost surely 3 (14.83%), and then comes positive probability for each integer (13.4%), almost surely 5 (5.26%), almost surely 6 (2.87%), and almost surely 4 (2.39%).

Almost surely 2 (briefly AS2) and almost surely infinity (ASI) can form a government with no need for a larger coalition. But they represent two political extremes. Is AS3 politically closer to AS2 or to ASI? “k with probability p_k for every k>2” (briefly, COM) represent a complicated political massage. Is it closer to AS2 or to ASI? (See the old posts on which coalition will be formed.)

**TYI37 poll: Partial results. It was exciting to see how the standing of the answers changed in the process of counting the votes.**

And the correct answer is:

See the paper:

Itai Benjamini and Gady Kozma: Arithmetic progressions in the trace of Brownian motion in space

Update: See Yuval Peres’ comment with an intuitive explanation.

]]>If we count **rooted** *k*-trees where the root is a *k*-clique the formula becomes somewhat simpler.

In 1972, when I was a teenage undergraduate student I was very interested in various extensions of Cayley’s formula for counting labeled trees. I thought about the question of finding a Prüfer code for* k*-trees and how to extend the results by Beineke and Pippert when for every clique of size in the *k*-tree we specify its “degree”, namely, the number of *k*-cliques containing it. (I will come back to the mathematics at the end of the post.) I thank Miki Simonovits for the photos and description and very helpful comments.

**Above, Kató Renyi, Paul Turan, Vera Sós, and Paul Erdős ; below Kató, Vera, and Lea Schönheim. Pictures: Jochanan (Janos) Schönheim.**

**From right, Rényi, Tur án and Erdős and Grätzer. **

While I was working on enumeration of -trees I came across a paper by Catherine Rényi and Alfréd Rényi that did everything I intended to do and quite a bit more.

What caught my eye was a heartbreaking footnote: when the paper was completed Catherine Rényi was no longer alive.

The proceedings where the paper appeared were of a conference in combinatorics in Hungary in 1969. This was the first international conference in combinatorics that took place in Hungary. The list of speakers consists of the best combinatorialists in the world and many young people including Laci Lovasz, Laci Babai, Endre Szemeredi, and many more who since then have become world-class scientists.

Years later Vera Sós told me the story of Alfréd Rényi’s lecture at this conference, the first international conference in combinatorics that took place in Hungary: “Kató died on August 23, on the day of arrival of the conference on “Combinatorial Theory and its Applications” (Balatonfured, August 24-29). Alfréd Renyi gave his talk (with the same title as the paper) on August 27 and his talk was longer than initially scheduled. They proved the results in the paper just the week before the conference. The paper appeared in the proceedings of the conference.”

Alfréd Renyi was one of the organizers of the conference and also served as one of the editors of the proceedings of the conference, which appeared in 1970. A few months after the conference, on February 1, 1970 Alfréd Rényi died of a violent illness. The proceedings are dedicated to the memory of Catherine Rényi and Alfréd Rényi.

Two pictures showing Alfréd and Catherine Rényi and a picture of Alfred Rényi and Paul Erdős.

Repeating a picture from last-week post. From left: Sándor Szalai, Catherine Rényi, Alfréd Rényi, András Hajnal and Paul Erdős (Matrahaza)

Going back to my story. I was 17 at the time and naturally I wondered if counting trees and similar things is what I want to do in my life. Shortly afterwards I went to the army. Without belittling the excitement of the army I quickly reached the conclusion that I prefer to count trees and to do similar things. My first result as a PhD student was another high dimensional extension of Cayley’s formula (mentioned in this post and a few subsequent posts). The question of how to generalize both formulas for -trees and for my hypertrees is still an open problem. We know the objects we want to count, we know what the outcome should be, and we know that we can cheat and use weighted counting, but still I don’t know how to make it work.

Some more comments on k-trees:

- Regarding the degree sequences for k-trees. You cannot specify the actual (k-2)-faces because those (in fact just the graph) determines the k-tree completely. So you need to count rooted k-trees and to specify the (k-2)-faces in terms of how they “grew” from the root.
- The case that all degrees are 1 and 2 that correspond to paths for ordinary trees and to triangulating polygons with diagonals for 2-trees are precisely the stacked (k-1)-dimensional polytopes. This is a special case of the Renyi & Renyi formula that was also found, with a different proof, by Beineke and Pippert.
- It is unlikely that there would be a matrix-tree formula for k-trees since telling if a graph contains a 2-tree is known to be NP complete. See this MO question. Maybe some matrix-tree formulas are available when we start with special classes of graphs.
- Regarding the general objects – those are simplicial complexes that are Cohen-Macaulay and their dual (blocker) is also Cohen-Macaulay.

This post is just about a single paper of Catherine Rényi and Alfréd Rényi mainly through my eyes from 45 years ago. Catherine Rényi’s main interest originally was Number theory, she was a student of Turàn, and soon she became interested in the theory of Complex Analytic Functions. Alfréd Rényi was a student of Frigyes Riesz and he is known for many contributions in number theory, graph theory and combinatorics and primarily in probability theory. Alfréd Rényi wrote several papers about enumeration of trees, and this joint paper was Catherine Rényi ‘s first paper on this topic.

]]>Let me draw your attention to one question raised and discussed in one of the discussion sessions.

Now, lets skip a lot of interesting staff and move to minute 23:20 where Noga Alon asked Misha Gromov to elaborate a statement from his opening lecture of the conference that the densest packing problem in is not interesting. In what follows Misha Gromov passionately argued that natural problems are bad problems (or are even stupid questions), and a lovely discussion emerged (in 25:00 Yuval Neeman commented about cosmology in response to Connes’s earlier remarks but then around 27:00 Vitali asked Misha to name some bad problems in geometry and the discussion resumed.) Misha made several lovely provocative further comments: he rejected the claim that this is a matter of taste, and argued that people make conjectures when they absolutely have no right to do so.

** Misha argues passionately that natural problems are stupid problems**

Actually one problem that Misha mentioned in his lecture as interesting (see also Gromov’s proceedings paper Spaces and questions), and that was raised both by him and by me is to prove an exponential upper bound for the number of simplicial 3-spheres with n facets. I remember that we talked about it in the conference and Misha was certain that the problem could be solved for shellable spheres while I was confident that the case of shellable spheres would be as hard as the general case. He was right! This goes back to works of physicists Durhuus and Jonsson see this paper On locally constructible spheres and balls by Bruno Benedetti and Günter M. Ziegler.

encore

Vitali Milman attacked the solution of the 4CT as “bad”and Segei Novikov disagreed and referred to the proof as “great”.

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The proceedings appeared as GAFA special volumes, but alas the articles are not electronically available even to GAFA’s subscribers. Let me encourage both Birkhauser and the contributors to make them more available. My talk was: An invitation to Tverberg’s theorem, and my own contribution to the Proceedings Combinatorics with a Geometric Flavor is probably the widest scope survey article I ever wrote. At the end of each section I added a brief philosophical thought about mathematics and those are collected in the post “about mathematics“.

Two more things: The conference (and few others organized by Vitali) was in Tel Aviv with a few days at the dead see and this worked very nicely. Vitali also organized in the mid 90s another very successful geometry conference unofficially celebrating Gromov’s 50th birthday with, among others, a very nice lecture by Gregory Perelman. If videos will become available I will be delighted to invite you to that conference as well. Update from Vitali: It was also the week of Jeff Cheeger’s 50th birthday which was also celebrated. Grisha Perelman gave an absolutely excellent talk talk on works of Cheeger. Lectures were not videotaped.

Avi Wigderson’s lecture

Are so called “natural questions” good for mathematics. Specifically is Kepler’s questions about the densest packing of unit balls in 3-space interesting? Watch a discussion of Misha Gromov, Noga Alon, Laci Lovasz and others. (next post)

We were all so much younger! (And in that old millennium, we were also all men )

Misha Gromov argues passionately that natural problems are bad problems (see next post)

Pictures of most participants ad two slides are below

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Spoiler: In one of the most intense scenes, the protagonist, with his bare hands and against all odds, took care of the mighty Wigner semi-circle law in two different ways. (From VV’s FB)

More information on Van Vu’s series of lectures. Van Vu’s home page; Related posts: did physicists really just prove that the universe is not a computer simulation—that we can’t be living in the Matrix? (Shtetl-Optimized); A related 2012 post on What’s New;

Two more pictures the first also from FB

Shaharon Shelah ICM 1974 (Vancouver) (Mohammad Golshani over FB)

See my post A theorem about infinite cardinals everybody should know; Gowers’s post Two infinities that are surprisingly equal about a recent breakthrough result by Maryanthe Malliaris and Saharon Shelah; and a recent 4-page solution to a conjecture of Spencer on finitary Hindman numbers by Shahram Mohsenipour and Shelah. More information on the last paper: (1) It is an Iranian-Israeli collaboration (2) Spencer asked Shelah the question during the workshop: Combinatorics: Challenges and Applications, celebrating Noga Alon’s 60th birthday, Tel Aviv University, January 17-21, 2016. (3) This is paper 1146 in Shelah’s (main) list of publications. Shelah’s 1974 lecture was called “Why There Are Many Nonisomorphic Models for Unsuperstable Theories.”

From left: Sándor Szalai, Catherine Rényi, Alfréd Rényi, András Hajnal and Paul Erdős (Matrahaza ) The picture is from Janos Pach’s Lancaster lecture, who also discussed how Szalai came up with Ramsey’s theorem. (See also Noga Alon and Michel Krivelevich’s chapter Extremal and Probabilistic Combinatorics, In: Princeton Companion to Mathematics, W. T. Gowers, Ed., Princeton University Press 2008, pp. 562-575.)

In the course of an examination of friendship between children some fifty years ago, the Hungarian sociologist Sandor Szalai observed that among any group of about twenty children he checked he could always find four children any two of whom were friends, or else four children no two of whom were friends. Despite the temptation to try to draw sociological conclusions, Szalai realized that this might well be a mathematical phenomenon rather than sociological one. He got interested in the problem, discussed it with Erdős, Rényi , and Turán and in a short time he came up with a number of interesting constructions. In fact, he obtained record lower bound estimates for several Ramsey numbers.

(Janos’ further remarks: “Sandor (Alexander) Szalai was a well known Hungarian sociologist and a famously bright and witty man. I am not sure whether he was the first to notice and study the the laws of clique- and anti-clique formation among groups of schoolchildren, but I suspect that he was not. The sociology of small groups used to be a popular alternative to more “dangerous” Marxist theories of classes in the 50-ies and 60-ies. My guess would be that Szalai discussed these issues and was fascinated by this subject some time around 1960. It is fair to say that he independently* conjectured* Ramsey’s theorem.”)