The Ramsey number R(s, t) is defined to be the smallest n such that every graph of order n contains either a clique of s vertices or an independent set of t vertices. Understanding the values of R(s,t) is among the most important, fruitful, and frustrating questions in combinatorics.

43 ≤ R(5,5) ≤ 48

Vigleik Angeltveit and Brendan D. McKay proved that the value of the diagonal Ramsey number R(5,5) is at most 48. Here is the paper.

The proof of is via computer verification, checking approximately two trillion separate cases! Congratulations Brendan and Vigleik! (I heard about it from Greg Kuperberg.)

The best known lower bound of 42 was established by Exoo in 1989. The previous best upper bound of 49 was proved by Brendan McKay and Stanislaw P. Radziszowski.

By this 4-days old theorem we now have 43 ≤ R(5, 5) ≤ 48. Brendan and Vigleik write “The actual value of R(5, 5) is widely believed to be 43, because a lot of computer resources have been expended in an unsuccessful attempt to construct a Ramsey (5,5)-graph of order 43.”

R(4,5) =25

In 1995 Brendan McKay and Stanislaw Radziszowski famously proved that R(4, 5) = 25. Listing all graphs with 24 vertices which admit a coloring without a clique on 5 vertices or an independent set with 4 vertices is a major part of the current proof.

Here is, more or less, how Greg described the achievement on FB: How many people can you have at a party such that no 5 are all acquainted and no 5 are all strangers? It has been known for nearly 30 years that 42 people is possible, and known for 20 years that 49 is not possible. The momentous news is that 48 is also not possible.

Greg continuation was a little cryptic to me: According to the paper, most mathematicians with any opinion on the matter agree with Douglas Adams that 42 is the answer. (By convention, the question is phrased in terms of the first impossible value, which is now known to be at least 43 and at most 48, and conjectured to be the former.) In response to my confusion Barry Simon added a ring theoretic aspect: Gil Kalai Isn’t the answer to “the ring of what” always: “one ring to rule them all” sort of like what “42” always means.

On the occasion of Polymath 12 devoted to the Rota basis conjecture let me remind you about the Alon-Tarsi conjecture and test your intuition concerning a strong form of the conjecture.

The sign of a Latin square is the product of signs of rows (considered as permutations) and the signs of columns. A Latin square is even if its sign is 1, and odd if its sign is -1. It is easy to see that when is odd the numbers of even and odd Latin squares are the same.

The Alon-Tarsi Conjecture: When is even the number of even Latin square is different from the number of odd Latin square.

This conjecture was proved when and is prime by Drisko and when and is prime by Glynn. The first open case is .

A stronger conjecture that is supported by known data is that when is even there are actually more even Latin squares than odd Latin squares. (This table is taken from the Wolfarm mathworld page on the conjecture.)

Test your intuition: Are there more even Latin squares than odd Latin squares when is even?

A poll

Unlike previous “test your intuition” questions the answer is not known.

The Alon Tarsi-Theorem

The Alon-Tarsi conjecture arose in the context of coloring graphs from lists. Alon and Tarsi proved a general theorem regarding coloring graphs when every vertex has a list of colors and the conjecture comes from applying the general theorem to Dinits’ conjecture that can be regarded as a statement about list coloring of the complete bipartite graph . In 1994 Galvin proved the Dinitz conjecture by direct combinatorial proof. See this post. Gian-Carlo Rota and Rosa Huang proved that the Alon-Tarsi conjecture implies the Rota basis conjecture (over ) when is even.

Let be a graph on vertices . Associate to every vertex a variable . Consider the graph polynomial Alon and Tarsi considered the coefficient of the monomial . If this coefficient is non-zero then they showed that for every lists of colors, colors for vertex , there is a legal coloring of the vertices from the lists! Alon and Tarsi went on to describe combinatorially the coefficient as the difference between numbers of even and odd Euler orientations.

Let me mention the paper by Jeannette Janssen The Dinitz problem solved for rectangles that contains a proof based on Alon-Tarsi theorem of a rectangle case of Dinitz conjecture. It is very interesting if the polynomial used by Janssen can be used to prove a “rectangular” (thus a bit weaker) version of the Rota’s basis conjecture. A similar slightly weaker form of the conclusion of Rota’s basis conjecture is conjectured by Ron Aharoni and Eli Berger in much much greater generality.

This is first of three posts kindly written by Thilo Weinert

Recently Gil asked me whether I would like to contribute to his blog and I am happy to do so. I enjoy both finite and infinite combinatorics and it seems that these fields drifted apart in recent decades. The latter is a comment not possibly substantiated by personal experience yet rather by my perception of the number and kinds of papers published. I believe that there is no mathematical reason for this separation but that it came about for reasons of the sociology of mathematics—maybe one should create a chair for this subject somewhere sometimes. Anyhow, I would like to write a few words about an area where finite and infinite combinatorics come together, the partition calculus for ordinal numbers.

The Partition Calculus, contemporarily classified as 03E02 under Mathematical Logic/Set Theory was initiated in 1956 with the article A partition calculus in set theory, published by Paul Erdos and Richard Rado in the Bulletin of the American Mathematical Society. asserts that for every set of -sized subsets of a set of size there is an of size such that all subsets of of size are in or an of size such that no subset of of size is in . Many mathematicians are concerned with the case where are finite. As in the finite realm the notions of cardinal number and ordinal number coincide, it is there unnecessary to differentiate between these notions of size. Furthermore, via the Well-Ordering-Theorem, the Axiom of Choice implies that for cardinalities the statement is equivalent to where denotes the smallest ordinal number of cardinality . It also implies that there is no such that . Hence, believing the Axiom of Choice, one may limit ones study to the cases in which are order-types and is a natural number. Although there are interesting open questions in contexts where the Axiom of Choice fails and also in contexts where not all of are ordinal numbers, I will for now exclude these from the discussion. That is, what I would like to talk about is the subject of transfinite Ramsey Numbers, is to say that but for no .

First I would like to discuss the lower storeys of the transfinite. It is known that generally for a countable ordinal and a natural number the Ramsey number is countable (If no subscript appears it is understood to be ). When is a finite multiple of or , the number’s calculation is similar in character to the case where is a natural number though it tends to be slightly more involved. For example where is the least number without a digraph on vertices without an independent -tuple and without an induced transitive subtournament on vertices. The are the Tournament Ramsey Numbers which have been investigated slightly more intensely in [1], [2] and exact values are known for . The last paper on these problems was published in 1997 by Jean Larson and William Mitchell in the very first issue of the Annals of Combinatorics. A degree argument yields which gives a good idea about the growth rate of as counterexamples to numbers being finite Ramsey numbers easily carry over. Furthermore they found a digraph on thirteen vertices without a transitive triple or independent quadruple thus establishing .

Recently I started to discuss these problems with Ferdinand Ihringer and Deepak Rajendraprasad. The latter found a digraph with the two aforementioned properties but on fourteen vertices and shortly thereafter they could establish by a triangle-count that there is no such digraph on vertices. So we know now that . Generally these problems should provide a nice playground for people in the business of solving Ramsey-type problems with the help of computers, cf. [3].

The situation is slightly more complicated in the case of finite multiples of as a degree argument only yields a cubic upper bound for which is a number defined such that for all natural numbers and . This is elaborated on in detail in [4].

Recently Jacob Hilton has considered problems of this kind involving additional topological structure alone (cf. [5]) and together with Andr\'{e}s Caicedo (cf. [6]).

In the next post I am going to elaborate on results and open questions
regarding Ramsey numbers for larger countable ordinals.

[1] Kenneth Brooks Reid, Jr. and Ernest Tilden Parker. Disproof of a conjecture of
Erdos and Moser on tournaments. J. Combinatorial Theory, 9 (1970).

[2] Adolfo Sanchez-Flores. On tournaments and their largest transitive subtournaments.
Graphs Combin., 10, 1994.

[3] Stanislaw P. Radziszowski. Small Ramsey numbers. Electron. J. Combin., 1:Dynamic
Survey 1, 30 pp. (electronic), 1994,

Let me mention a strong version of Rota’s conjecture (Conjecture 6 in that paper) due to Jeff Kahn that asserts that if you have bases of an -dimensional vector space then you can choose such that for every , the vectors form a basis of and also the vectors form a basis. This conjecture can be regarded as a strengthening of the Dinits Conjecture whose solution by Galvin is described in this post. Rota’s original conjecture is the case where depends only on .

A very nice special case of Rota’s conjecture was proposed by Jordan Ellenberg in the polymath12 thread: Given trees on vertices, is it always possible to order the edges of each tree so that the th edges in the trees form a tree for every ?

I have some plans to write about the partitionability conjecture (and an even more general conjecture of mine) soon. But now I would like to draw your attention to a weakening of these conjectures, still implying the “numerical” consequences of the original conjectures, that was proved in 2000 by Art Duval and Ping Zhang in their paper Iterated homology and decompositions of simplicial complexes . The partitionability conjecture is about decompositions into subcubes (or intervals in the Boolean lattice) and the result is about decomposition into subtrees of the Boolean lattice. (See here for the massage “trees not cubes!” in another context.)

Two steps from fame

This pictures was taken by Edna Wigderson in a 50th birthday party for Avi Wigderson. Unfortunately the picture shows us while landing from our 3 meter high (3.28 yards) jumps and thus does not fully capture the achievement.

The guy on the left is Bernard Chazelle, a great computer scientist and geometer, a long time friend of me and Avi, and the father of Damien Chazelle the director and writer of the movie La La Land now nominated for the record 14 Oscar awards. I wish Damien to win a record number of Oscars and to continue writing, directing, and producing wonderful movies so as to keep shattering his own records and giving excitement and joy to hundreds of millions, perhaps even billions, of people. (Update: Six Oscars, Damien the youngest ever director to win.)

The Jerusalem Baroque orchestra and 2017 Bach Festival

For those in Israel let me draw your attention to the Jerusalem Baroque orchestra and especially to the 2017 Bach Festival. (I thanks Menachem Magidor for telling me about this wonderful orchestra.)

Theorem (Hopf and Pannwitz, 1934): Let be a set of points in the plane in general position (no three points on a line) and consider line segments whose endpoints are in . Then there are two disjoint line segments.

Micha Perles’s proof by Lice:

Useful properties of lice: A louse lives on a head and wishes to lay an egg on a hair.

Think about the points in the plane as little heads, and think about each line segments between two points as a hair.

The proof goes as follows:

Step one: You take lice from your own head and put them on the points of $X$.

Step two: each louse examines the hairs coming from the head and lay eggs (on the hair near the head)

Step three (not strictly needed): You take back the lice and put them back on your head.

To make it work we need a special type of lice: spoiled-left-wing-louse.

Three children inherited 17 camels. The will gave one half to one child, one third to a second child and one ninth to the third. The children did not know what to do and a neighbor offered to lend them a camel. Now there were 18 camels. One child got camels the second got camels, and the third got camels. Altogether they got 17 camels so they could give back the extra camel to the kind neighbor.

Theorem: A bipartite cubic graph has a perfect matching.

(A cubic graph is a 3-regular graph.)

Proof: Suppose that has vertices. Multiply each edge times ( large) so that the degree of each vertex is of the form . Now ask your neighbor to give you an additional perfect matching and add it to the graph which now is regular of degree . The new graph has an Eulerian cycle. (If not connected, every connected component has an Eulerian cycle.) When we walk on the Eulerian cycle and take either the even edges or the odd edges we get two subraphs of degree . At least one of them does not use all the edges of the neighbor. We move to this subgraph and give the unused edge back to the neighbor. We repeat, and in each step we move to a regular subgraph of degree a smaller power of two, and give back at least one edge to the kind neighbor. If is large enough to start with we will end with a perfect matching that uses only the original edges of our graph.

(Remark: We can take or . If we are a little more careful and in each step try to give many edges back to the kind neighbor we can use or so.)

Some personal/historical remarks in first minutes of my lecture at 7ECM on July 2016…

German-Jewish mathematicians in the early days of the Hebrew University of Jerusalem

Being invited to give a plenary lecture at the 7ECM was a great honor and, as Keren Vogtmann said in her beautiful opening lecture on outer spaces, it was also a daunting task. I am thankful to Günter Ziegler for his introduction. When I ask myself in what way I am connected to the person I was thirty years ago, one answer is that it is my long-term friendship with Günter and other people that makes me the same person. My lecture deals with the analysis of Boolean functions in relation to expansion (isoperimetric) properties of subsets of the discrete n-dimensional cube. The lecture has made a subjective selection of some results, proofs, and problems from this area.

Yesterday, Leonid Polterovich and I were guests of the exhibition “Transcending Tradition: Jewish Mathematicians in German-Speaking Academic Culture.” I will start by briefly mentioning the great impact of German-Jewish mathematicians on the early history of the Einstein Institute of Mathematics and Physics at the Hebrew University of Jerusalem, my main academic home since the early seventies. In this picture you can see some early faces of our Institute.

Edmund Landau, the founder and first head of the Institute, moved to Jerusalem from Göttingen in 1927 and moved back to Göttingen a year and a half later. Abraham (Adolf) Halevi Fraenkel moved to Jerusalem from Kiel in 1928 and he can be seen as the father of logic, set theory, and computer science in Israel. My own academic great-grandfather is Michael Fekete, who immigrated to Jerusalem from Budapest in 1928.

Two remarkable documents by Edmund Landau

I would like to say a few words about two remarkable documents written by Landau in 1925, both related to the inauguration ceremony of the Hebrew University of Jerusalem. You can read more about them in the paper Zionist internationalism through number theory: Edmund Landau at the Opening of the Hebrew University in 1925 by Leo Corry and Norbert Schappacher . The first document is Landau’s toast for the opening ceremonies. Let me quote two sentences:

May great benefit emerge from this house dedicated to pure science, which does not know borders between people and people. And may this awareness emerge from Zion and penetrate the hearts of all those who are still far from this view.

The second document, also from 1925, is probably the first mathematical paper written in Hebrew in modern times. It is devoted to twenty-three problems in number theory and here are its concluding sentences.

At this number of twenty-three problems I want to stop, because
twenty-three is a prime number, i.e., a very handsome number for us. I am certain that I should not fear to be asked by you, for what purpose does one deal with the theory of numbers and what applications may it have. For we deal with science for the sake of it, and dealing with it was a solace in the days of internal and external war that as Jews and as Germans we fought and still fight today.

I wish to make two remarks: First, note that Landau moved from the very ambitious hopes and program of science as a bridge that eliminates borders between nations to a more modest and realistic hope that science and mathematics give comfort in difficult times. Juggling between very ambitious programs and sober reality is in the nature of our profession and we are getting paid both for the high hopes and aims, as well as for the modest results. Second, Landau is famous for his very rigorous and formal mathematical style but his 1925 lecture is entertaining and playful. I don’t know if his move to Jerusalem was the reason for this apparent change of style. Parts of Landau’s lecture almost read like stand-up comedy. Here is, word for word, what Landau wrote about the twin prime conjecture:

Satan knows [the answer]. What I mean is that besides God Almighty no one knows the answer, not even my friend Hardy in Oxford.

These days, ninety years after Landau’s lecture, we can say that besides God Almighty no one knows the answer and not even our friend James Maynard from Oxford. We can only hope that the situation will change before long.

Landau’s hopeful comments were made only nine years after the end of the terrible First World War. He himself died in 1938 in Berlin, after having been stripped of his teaching privileges a few years earlier. I don’t know to what extent the beauty of mathematics was a source of comfort in his last years, but we can assume that this was indeed the case. My life, like the lives of many others of my generation, was overshadowed by the Second World War and the Holocaust and influenced by the quest to come to terms with those horrible events.

Trying to follow an example of a 1925 lecture by Landau (mentioned in the lecture), the writing style is very much that of a lecture. It goes without saying that I will be very happy for corrections and suggestions of all kinds.

I am sure that every one of the readers of this blog heard about Laci Babai’s quasi-polynomial algorithm for graph isomorphism and also the recent drama about it: A mistake pointed out by Harald Helfgott, a new sub-exponential but not quasi-polynomial version of the algorithm that Laci found in a couple of days, and then, a week later, a new variant of the algorithm again found by Laci which is quasi-polynomial. You can read the announcement on Babai’s homepage, three excellent Quanta magazine articles by Erica Klarreich(I,II,III), Blog posts over Harald’s blog (III,II,I) with links to the video and article (in French), and many blog posts all over the Internet (GLL4,GLL3,GLL2,GLL1,…).

Babai’s result is an off-scale scientific achievement, it is wonderful in many respects, and I truly admire and envy Laci for this amazing breakthrough. I also truly admire Harald for his superb job as a Bourbaki expositor.

Laci Babai is visiting and he is giving lectures on graph isomorphism and related topics all over the Israel.

Tel Aviv University

Tel Aviv University: Sackler distinguished lectures in Pure Mathematics Wednesday, January 18 (Poster. Sorry, too late, I heard it was very inspiring, don’t miss the other talks!)

Tel Aviv University Combinatorics seminar: Sunday, Jan. 22, 10:00-11:00, Location: Melamed (Shenkar building, ground floor, room 6)
Title: Canonical partitioning and the emergence of the Johnson graphs:Combinatorial aspects of the Graph Isomorphism problem

(The talk does not depend on Wednesday’s talk)

Hebrew University of Jerusalem

Hebrew University Colloquium San. Jan 22, 16:00-17:00 Title: Graph isomorphism and coherent configurations: The Split-or-Johnson routine

Lecture room 2, Manchester building (Mathematics)

The Technion

Local versus global symmetry and the Graph Isomorphism problem I–III

Lecture I: Monday, January 23, 2017 at 15:30

Lecture II: Tuesday, January 24, 2017 at 15:30

Lecture III: Thursday, January 26, 2017 at 15:30

All lectures will take place at Auditorium 232, Amado Mathematics Building, Technion (Website)

Weitzman Institute

Pekeris lecture, Jan 29, 11:00-12:00 Hidden irregularity versus hidden symmetry

The Polymath10 project on the Erdos-Rado Delta-System conjecture took place over this blog from November 2015 to May 2016. I aimed for an easy-going project that people could participate calmly aside from their main research efforts and the duration of the project was planned for one year. I also wanted to propose and develop my own homological approach to the problem.

The purpose of this post is to (belatedly) formally announce that the project has ended, to give links to the individual posts and to briefly mention some advances and some thoughts about it.

The problem was not solved and we did not come near a solution. The posts contain some summary of the discussions, a few results, and some proposals by the participants. Phillip Gibbs found a remarkable relation between the general case and the balanced case. Dömötör Palvolgyi shot down quite a few conjectures I made, and Ferdinand Ihringer presented results about some Erdos-Ko-Rado extensions we considered (In term of upper bounds for sunflower-free families). Several participants have made interesting proposals for attacking the problem.

I presented in the second post a detailed homological approach, and developed it further in the later threads with the help of Eran Nevo and a few others. Then, after a major ingredient was shot down, I revised it drastically in the last post.

Participants made several computer experiments, for sunflower-free sets, for random sunflower-free sets, and also regarding the homologica/algebraic ideas.

The posts (and some comments) give some useful links to literature regarding the problem, and post 5 was devoted to a startling development which occurred separately – the solution of the Erdos-Szemeredi sunflower conjecture for sunflowers with three petals following the cup set developments. (The Erdos-Szemeredi sunflower conjecture is weaker than the Erdos-Rado conjecture.)

The origin of my homological approach

A (too) strong version of the homological conjecture appeared in my 1983 Ph. D. thesis written in Hebrew. The typesetting used the Hebrew version of Troff.