Infinite Structural Ramsey Theory (25w5398)


Jan Hubicka (Charles University)

Natasha Dobrinen (University of Notre Dame)

Stevo Todorcevic (University of Toronto)

Andy Zucker (University of Waterloo)


The Banff International Research Station will host the “Infinite structural Ramsey theory” workshop in Banff from November 23 to 28, 2025.

The pigeon-hole principle asserts that if $n+1$ pigeons attempt to live in only $n$ nests, then some two pigeons must share. This fundamental principle and its numerous and powerful generalizations comprise the field of mathematics known today as Ramsey theory. This workshop will focus on \emph{structural} Ramsey theory, the study of Ramsey-theoretic principles on classes of first-order structures, such as graphs (Abramson-Harringon; Ne\v{s}et\v{r}il-R\"odl), Boolean algebras (Graham-Rothschild), or the class of all finite structures in any language, even possibly containing functions (Hubi\v{c}ka-Ne\v{s}et\v{r}il). Recently, much progress has been made in infinite structural Ramsey theory, in particular the study of big Ramsey degrees. Starting with work of Dobrinen on triangle-free and $K_n$-free graphs, the area has seen a resurgence, including work of Zucker on more general binary free amalgamatation classes, alternate proof techniques introduced by Hubi\v{c}ka, and precise characterizations of big Ramsey degrees for numerous classes by Balko, Chodounsky, Dobrinen, Hubi\v{c}ka, Kone\v{c}n\'y, Vena, and Zucker. This field of research has matured to the point where deep connections with other areas of mathematics, such as topological dynamics, computability theory, and functional analysis are ready to be explored fully.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada’s Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), and Alberta’s Advanced Education and Technology.