Natasha Dobrinen (University of Denver)

Ramsey theory on infinite structures

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The Infinite Ramsey Theorem says that for any positive integer $n$, given a coloring of all $n$-element subsets of the natural numbers into finitely many colors, there is an infinite set $M$ of natural numbers such that all $n$-element subsets of $M$ have the same color. Infinite Structural Ramsey Theory is concerned with finding analogues of the Infinite Ramsey Theorem for Fraïssé limits, and also more generally for universal structures. In most cases, the exact analogue of Ramsey's Theorem fails. However, sometimes one can find bounds of the following sort: Given a finite substructure $A$ of an infinite structure $S$, we let $T(A,S)$ denote the least number, if it exists, such that for any coloring of the copies of $A$ in $S$ into finitely many colors, there is a substructure $S'$ of $S$, isomorphic to $S$, such that the copies of $A$ in $S'$ take no more than $T(A,S)$ colors. If for each finite substructure $A$ of $S$, this number $T(A,S)$ exists, then we say that $S$ hasfinite big Ramsey degrees.

In the past six years, there has been a resurgence of investigations into the existence and characterization of big Ramsey degrees for infinite structures, leading to many new and exciting results and methods. We will present an overview of the area and some highlights of recent work by various author combinations from among Balko, Barbosa, Chodounsky, Coulson, Dobrinen, Hubicka, Konjecny, Masulovic, Nesetril, Patel, Vena, and Zucker.