Thilo Weinert: Transfinite Ramsey Numbers

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. $\kappa \rightarrow (\lambda, \mu)^\nu$ asserts that for every set $N$ of $\nu$-sized subsets of a set $K$ of size $\kappa$ there is an $L \subset K$ of size $\lambda$ such that all subsets of $L$ of size $\nu$ are in $N$ or an $M \subset K$ of size $\mu$ such that no subset of $M$ of size $\nu$ is in $N$. Many mathematicians are concerned with the case where $\kappa, \lambda, \mu, \nu$ 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 $\kappa,\lambda,\mu$ the statement $\kappa\rightarrow(\lambda,\mu)^\nu$ is equivalent to $\bar{\kappa}\rightarrow(\bar{\lambda}, \bar{\mu})^\nu$ where $\bar{\zeta}$ denotes the smallest ordinal number of cardinality $\zeta$. It also implies that there is no $\kappa$ such that $\kappa\rightarrow(\omega, \omega)^\omega$. Hence, believing the Axiom of Choice, one may limit ones study to the cases in which $\kappa,\lambda,\mu$ are order-types and $\nu$ 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 $\kappa,\lambda,\mu$ 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, $\kappa=r_\nu(\lambda,\mu)$ is to say that $\kappa\rightarrow(\lambda, \mu)^\nu$ but $\zeta\rightarrow(\lambda,\mu)^\nu$ for no $\zeta<\kappa$.

First I would like to discuss the lower storeys of the transfinite. It is known that generally for a countable ordinal $\lambda$ and a natural number $\mu$ the Ramsey number $r_2(\lambda, \mu)$ is countable (If no subscript appears it is understood to be $2$). When $\lambda$ is a finite multiple of $\omega$ or $\omega^2$, the number’s calculation is similar in character to the case where $\lambda$ is a natural number though it tends to be slightly more involved. For example $r_2(\omega\ell, m)=\omega r(I_\ell, L_m)$ where $r(I_\ell, L_m)$ is the least number $n$ without a digraph on $n$ vertices without an independent $\ell$-tuple and without an induced transitive subtournament on $m$ vertices. The $r(I_2, L_n)'s$ are the Tournament Ramsey Numbers which have been investigated slightly more intensely in [1], [2] and exact values are known for $\ell \leqslant 6$. 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 $r(I_\ell,L_3)\leq\ell^2$ which gives a good idea about the growth rate of $r(I_\ell, L_3)$ 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 $r(I_\ell, L_3) \in \{14, 15, 16\}$.

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 $15$ vertices. So we know now that $r(I_4, L_3) = 15$. 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 $\omega^2$ as a degree argument only yields a cubic upper bound for $r(I_\ell,A_3)$ which is a number defined such that $r(\omega^2\ell,m)=\omega^2r(I_\ell,A_m)$ for all natural numbers $\ell$ and $m$. 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,

[4] Thilo Volker Weinert, Idiosynchromatic poetry. Combinatorica, 34 (2014).

[5] Andres Eduardo Caicedo and Jacob Hilton, Topological Ramsey numbers and countable ordinals.

[6] Jacob Hilton. The topological pigeonhole principle for ordinals.

Polymath12

Timothy Chow launched polymath12 devoted to the Rota Basis conjecture on the polymathblog. A classic paper on the subject is the 1989 paper by Rosa Huang and Gian Carlo-Rota.

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 $n^2$ bases $B_{i,j} 1 \le i \le n,1\le j \le n$ of an $n$-dimensional vector space then you can choose $b_ {ij} \in B_{i,j}$ such that for every $k,1 \le k\le n$, the vectors $b_{ik}:1\le i \le n$ form a basis of $V$ and also the vectors $b_{kj}:1\le j \le n$ 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 $B_{i,j}$ depends only on $i$.

A very nice special case of Rota’s conjecture was proposed by Jordan Ellenberg in the polymath12 thread: Given $n-1$ trees on $n$ vertices, is it always possible to order the edges of each tree so that the $k$th edges in the trees form a tree for every $k$?

Two recent papers in the Notices AMS

The February 2017 issue of the Notices of the AMS has two beautiful papers on topics we discussed here. Henry Cohn  wrote an article A conceptual breakthrough in sphere packing about the breakthrough on sphere packings in 8 and 24 dimensions  (see this post) and  Art Duval, Carly Klivans, and Jeremy Martin wrote an article on the the Partitionability Conjecture. (That we mentioned here.)

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.)

Proof By Lice!

From camels to lice. (A proof promised here.)

Theorem (Hopf and Pannwitz, 1934): Let $X$ be a set of $n$ points in the plane  in general position (no three points on a line) and consider $n+1$ line segments whose endpoints are in $X$.  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 $n$ lice from your own head and put them on the $n$ 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 $n$ lice and put them back on your head.

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

Posted in Combinatorics, What is Mathematics | Tagged | 5 Comments

The seventeen camels riddle, and Noga Alon’s camel proof and algorithms

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 $\frac{18}{2}=9$ camels the second got $\frac{18}{3}=6$ camels, and the third got $\frac{18}{9}=2$ camels. Altogether they got 17 camels so they could give back the extra camel to the kind neighbor.

In the very short 2003 paper A simple algorithm for edge-coloring bipartite multigraphs, Information Processing Letters 85 (2003), 301-302, Noga Alon used a similar idea for algorithmic purposes. (He also observed the connection to the camel riddle). Here is how the extra camel idea is used for:

Theorem:  A bipartite cubic graph $G$ has a perfect matching.

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

Proof: Suppose that $G$ has $n$ vertices. Multiply each edge $r$ times ($r$ large) so that the degree of each vertex $3r$ is of the form $2^m-1$. Now ask your neighbor to give you an additional perfect matching and add it to the graph which now is regular of degree $2^{m}$ . 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 $2^{m-1}$.  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 $r$ 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 $m=n/2$ or $m=n/2+1$. If we are a little more careful and in each step try to give many edges back to the kind neighbor we can use $m=\log n$ or so.)

Posted in Combinatorics | Tagged | 4 Comments

Edmund Landau and the Early Days of the Hebrew University of Jerusalem

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 Göttingen 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.

Here is the videotaped lecture. (and the slides).

More sources: The home page of my Institute, and a page about its history; An article in the  AMS Notices; A blog post in Hebrew about the 1925 Hebrew University events;  Shaul Katz: Berlin roots – Zionist incarnation: The ethos of pure mathematics and the beginnings of the Einstein Institute of Mathematics at the Hebrew University of JerusalemScience in Context 17(1/2), 199-234 (2004); A blog post about Yaakov Levitzki  and the Amitzur-Levitzki theorem; Schappacher, Norbert:  Edmund Landau’s Göttingen — From the life and death of a great mathematical center.   Mathematical Intelligencer 13 (1991), 12-18. (Talk at the Dedication of the Landau Center for Research in Mathematical Analysis, Jerusalem, Feb. 28th, 1989). A recent post on Tao’s blog related to mathematics, science, scientific relations and recent events.

(Below) First and last page of Hardy and Heilborn obituary on Landau.

Boolean Functions: Influence, Threshold, and Noise

Here is the written version of my address at the 7ECM last July in Berlin.

Boolean functions, Influence, threshold, and Noise

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.

Laci Babai Visits Israel!

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

EBNER AUDITORIUM (webpage)

Polymath10 conclusion

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 posts were

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.

Posted in Combinatorics, Open problems, Polymath10 | | 4 Comments

Five years ago I wrote a post entitled Is Backgammon in P? It was based on conversations with Peter Bro Miltersen and Uri Zwick (shown together in the above picture) about the computational complexity of computing the values (and equilibrium points) of various stochastic games, and also on some things I learned from my game theory friends over the years about proving that values exist for some related games. A few weeks ago  two former students of Peter, Rasmus Ibsen-Jensen and  Kristoffer Arnsfelt Hansen visited Israel and I had a chance to chat with them and learn about some recent exciting advances.

In what way is Backgammon harder than chess?

Is there a polynomial time algorithm for chess?  Well, if we consider the complexity of chess in terms of the board size then it is fair to think that the answer is “no”. But if we wish to consider the complexity in terms of the number of all possible positions then it is easy to go backward over all positions and determine  the outcome of the game when we start with each given position.

Now, what about backgammon?   Like chess, backgammon is a game of complete information.  The difference between backgammon and chess is the element of luck; at each position your possible moves are determined by a roll of two dice. This element of luck increases the computational  skill needed for playing backgammon compared to chess. It can easily be seen that optimal strategy for players in backgammon need not involve any randomness.

Problem 1: Is there a polynomial time algorithm to find the optimal strategy (and thus the value) of a stochastic zero sum game with perfect information? (Like backgammon)

This question (raised by Ann Condon in 1998) represents one of the most fundamental open problem in algorithmic game theory.

In what way is heads-up poker harder than backgammon?

Heads-up poker is just a poker game with two players.  To make it concrete you may think about heads-up Texas hold’em poker. This is not a game with complete information, but by according to the minmax theorem  it still has a value. The optimal strategies are mixed and involve randomness.

Problem 2: Is there a polynomial time algorithm to find the optimal strategy (and thus the value) of a stochastic zero-sum game with incomplete information? (like heads-up Texas hold’em poker).

It will be very nice to find even a sub-exponential algorithm for a stochastic zero-sum game with incomplete information like poker. Continue reading

The Median Game

Update: Apparently this game was invented already by Douglas Hofstadter who called it  “Mediocrity” and it is published in Hofstadter’s book Metamagical Themas: Questing for the Essence of Mind and Pattern. It is also called “Hruska.” (See here and here.)

Ehud Friedgut reminded me of the game MEDIAN which I proposed many years ago.

There are three players and they play the game for eight rounds. In every round all players simultaneously say a number between 1 and 8. A player whose number is (strictly) between the other two get a point. At the end of the game the winner is the player whose number of points is strictly between those of the others.

Posted in Games | Tagged | 4 Comments