# Unifying themes in Ramsey Theory – BIRS 2018

In November 2018, 41 of the top researchers in Ramsey theory met at the BIRS in Banff for the Unifying Themes in Ramsey Theory conference. By all measures the conference was a big success. What makes Ramsey theory so special is that it has wide ranging impacts in diverse fields in mathematics. The participants gave talks showing how Ramsey theory has impacted fields like graph theory, topological dynamics, set theory, model theory, operator algebras, logic and statistics.

Since I have a somewhat broad base of knowledge in Ramsey theory, I tried my best to give a short description of each of the speakers in language that makes sense to me. My view is biased, and my intent is always to show off the amazing work everyone is doing. I hope nothing comes across as negative or critical; that is not my intent.

You can find all the abstracts here, and all the videos of their talks here.

## The four must-watch presentations (IMHO)

Here are four talks I picked out that I think are important and worth watching for an outsider to Ramsey Theory.

## Monday – Day 1A – Combinatorics and Ramsey classes

To kick off the conference, Jaroslav Nesetril gave a nice overview of the types of Ramsey theorems, problems and perspectives we are likely to see this week. He told us about how Mendel (the biologist) was the first to use the $\binom{n}{k}$ notation [LINK]. Of the many important questions related to Ramsey theory, Jarik mentioned two:

• Are there other good tools for showing that a graph has high chromatic number?
• Does the class of graphs that forbid $C_4$ have a Ramsey expansion?

The second talk was supposed to be given by Michael Kompatscher, but he was unable to travel to Canada because of illness. So I (Mike Pawliuk) gave the second talk in his place, about connections to data science, big data, statistics and machine learning. Funny enough, I could have given Michael’s talk since it was about joint work that we did.

Matej Konecny gave the final talk of the morning, showcasing new results that capture the completion algorithms of many different classes of metric spaces. Completing large cycles to a complete metric graph is an important step in showing that a class has a Ramsey expansion. This technique has become more and more explicit in recent years.

## Monday – Day 1B – EPPA/Hrushovski property

Natasha Dobrinen started the afternoon with a chalkboard talk showcasing new results about big Ramsey degrees. She showed how she overcame the difficulties with the random graph to find the big Ramsey degrees for graphs that forbid small complete graphs. At its core she was able to sidestep a difficulty that the Sauer construction presented by starting where the Sauer construction ends.

Jan Hubicka highlighted exciting work that he’s been doing about general methods of proving the EPPA (Hrshovski property) for various classes of metric spaces. He was able to breathe life into the idea of valuations first presented elsewhere.

Finally, Macin Sabok presented work showing that the class of hypertournaments does not have EPPA. It uses generalizations of the ideas present in Herwig-Lascar, together with other ideas from algebraic topology. It does not answer the (very hard) prized problem “Does the class of tournaments have EPPA?”, but it does prove EPPA for a specific class of isomorphisms.

The day ended with a small group of us going on a fast hike of Tunnel mountain.

## Tuesday – Day 2A – Dynamics

The morning session focused on the fruitful connections that topological dynamics has had with Ramsey theory through the KPT correspondence.

Friedrich Martin Schneider gave the first talk which presented the “Gromov-Milman” perspective on the KPT correspondence. We saw a Gromov result that is a concentration of measure result, which corresponds to a Ramsey result through the KPT correspondence. Martin then showed us a stronger version of Gromov’s result, answering a question of Pestov.

Colin Jahel, a graduate student, gave an impressive talk about the semigeneric directed graph. It was especially interesting to me since it answered an open problem from my thesis that I was unable to solve. Colin presented results that reified techniques for proving unique ergodicity, and sidestepped using probabilistic arguments.

Andy Zucker rounded out the morning by giving an alternate perspective on the work of Colin (who is a coauthor). He discussed the dynamical perspective of the amenability and metrizable flows. This work is another step in Andy giving a very clear picture of what is happening with the universal minimal flows. This is one of the clearest, most straightforward talks about this topic I’ve ever seen.

From the audience, Stevo Todorcevic mentioned a nice (classical) result that if the product of two spaces contains a copy of $\beta \mathbb{N}$, then one of the factors must contain a copy of it. In this way, $\beta \mathbb{N}$ is an “irreducible” space.

## Tuesday – Day 2B – Applications of Ramsey theory

The afternoon session featured subtle uses of Ramsey theory underlying key theorems. This made many of the results possible, even if they didn’t directly invoke Ramsey results.

Wieslaw Kubis started the afternoon by presenting results about uniform homogenity, Katetov functors and mixed sums of Fraisse classes. The mixed sum construction is a type of “bipartite” construction where each part is a Fraisse structure. While he used this construction to provide a counterexample, it is also a broadly useful construction.

Milos Kurilis followed up with a proof of Vaught’s conjecture in the case of monomorphic functions. This result quietly uses the fact that chainable uses Ramsey type-results. It was a remarkably understandable talk (to me a non-expert in model theory), despite the technical nature of the material.

In his talk Milos produced one of the most beautiful diagrams I’ve ever seen in a math talk.

The final talk of the afternoon was David Hartman, who ended the day with a high-energy punch. David separated two closely related embedding properties with lots of examples and constructions. The final payoff for the day was a nice construction of the Rado graph partitioned into finitely many pieces.

After the presentations we had a problem session, where participants in the conference shared problems of interest. Here are the mostly-complete notes I took.

Ending the day, Wieslaw Kubis lead us in a problem session about the weak amalgamation property. About half the participants showed up and many people contributed to the discussion. Finally, at 8:30 (almost 12 hours after starting) we called it a day.

## Wednesday – Day 3A –

Wednesday contained only talks in the morning.

Martin Balko explained the project of ordered Ramsey numbers. He started by surveying the classical results about (usual) Ramsey numbers and contrasting them with the ordered versions. In many cases the bounds on the ordered/unordered Ramsey numbers are very different, even in the case of paths.

Lionel Nguyen Van Thé followed up by reminding us about Erdos-Rado type results about canonical colourings and equivalence relations. Big Ramsey results would become a recurring theme in this workshop. This talk stirred the most discussion about how it relates to canonical functions in other areas (like algebra).

The final talk of the morning was Sam Braunfeld, who gave an overview of the classification of homogeneous finite-dimensional permutation structures. Sam compared and contrasted his results with Cameron’s 2002 classification. This type of work is of particular interest to people in structural Ramsey theory, who use these catalogues as a source of interesting examples.

In the afternoon, some of us hiked up to Sundance Canyon.

## Thursday – Day 4A – Set Theory and Logic

Thursday’s schedule was anticipated to be rather heavy, but the speakers were very gentle to the audience and it ended up being one of the best days of the conference.

Martino Lupini started us off by explaining his intuition for his recent results about the generalized Tetris operations relating to Gower’s theorem. In particular he showed us how he used a perspective from non-standard analysis and the idempotent ultrafilter proof of Hindman’s theorem.

Francisco Guevara Parra gave a talk about Tukey orders and local Ramsey theory. This was one of the few talks to show the application of Ramsey theory to topological groups, and infinite combinatorics.

David Chodounsky ended the morning with a problem motivated by set theoretic forcing, but of independent interest to those studying Ramsey theory. David stirred up interest in his question about Halpern-Lauchi ideals. He also gave us a survey of the landscape of HL ideals including a very nice map of the known implications.

## Thursday – Day 4B

Jordi Lopez-Abad gave us a very nice presentation of various approximate Ramsey properties. He put special effort in to give examples and pictures and it was very appreciated. We saw that some of the studied objects were “shapes” where the boundary is kind of blurry, but you can still tell the difference between a square and a hexagon. This context is “almost Euclidean”. There was a lot of motivating intuition here.

Michael Pinsker described canonical functions but, funny enough, not the “canonical functions” he originally wanted to talk about! The motivation for his talk is a nearly complete paper he wrote in 2002 that contained a critical false lemma. This lemma is an (infinite) Ramsey theory problem, and Michael was hoping to spark some renewed interest in it.

Aleksandra (Ola) Kwiatkowska gave the final talk for the day, and it was one of the best of the conference. Ola showed that an obscure (but natural!) Fraisse class (the Weiewski dendrites) negatively answers two fundamental (and related) questions in the field (Does every omega-categorical structure have a precompact Ramsey expansion?). David Evans had given an answer to this in 2016, but Ola’s example is more natural, and provides a counterexample to some other conjectures as well.

After dinner, Michael Hrusak led a working session about Ramsey-type problems on Borel Ideals. You can watch the video of it.

## Friday – Day 5A – New Frameworks

The final day, like Wednesday, only had talks in the morning. The theme of the morning was new frameworks and directions for Ramsey theory.

Noé de Rancourt opened the talks with a discussion about Ramsey spaces, and specifically, what kind of results can you get if you don’t have a pigeonhole principle in that space. Ramsey spaces are very nice combinatorial objects that capture the essential Ramsey behaviour of many geometric objects; it is related to combinatorical forcing. Noé showed us how to relate these to games and local Ramsey theory.

Dragan Masulovic showed us how the tools of category theory can be used to view and prove dual results in Ramsey theory. He showed us the value and type of isomorphism of categories in the Ramsey context. There was a special attention to making the results usable and practical for the non-category theorist.

Stevo Todorcevic gave concluding remarks for the conference. To be honest, I was so caught up in his talk that I didn’t take notes. I highly recommend watching his talk.

And with that, we wrapped up this iteration of the Ramsey theory meeting.

## One thought on “Unifying themes in Ramsey Theory – BIRS 2018”

1. Thanks for the awesome write-up, Mike! Now I know where to start with the videos.

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