Ramsey theory of homogeneous structures: current trends and open problems

Abstract

This article highlights historical achievements in the partition theory of countable homogeneous relational structures, and presents recent work, current trends, and open problems. Exciting recent developments include new methods involving logic, topological Ramsey spaces, and category theory. The paper concentrates on big Ramsey degrees, presenting their essential structure where known and outlining areas for further development. Cognate areas, including infinite dimensional Ramsey theory of homogeneous structures and partition theory of uncountable structures, are also discussed.This article will appear in the 2022 ICM Proceedings. It is dedicated to Norbert Sauer for his seminal works on the partition theory of homogeneous structures, and for his mathematical and personal generosity.

Big Ramsey Degrees of the Generic Partial Order

“…Each level ℓ of this tree contains a special node type ℓ K (ℓ) called a coding node. Dobrinen [Dob20,Dob23] and Zucker [Zuc22] developed a Ramsey theorem for trees with coding nodes which is the main tool used in [Zuc22] (see also recent ICM survey [Dob21]). These trees are highly irregular and do not directly fit the framework of (S, M)-trees (even though a suitable (S, M)-tree can be constructed).…”

Infinite-dimensional Ramsey theory for binary free amalgamation classes

“…The big Ramsey degrees for these classes of structures were completely characterized in [2]; we will discuss this characterization in more detail later in the introduction. For further background and overview regarding the finite-dimensional Ramsey theory of countable structures, we refer the reader to [2] and [12].…”

Exact big Ramsey degrees via coding trees

“…Although work on big Ramsey degrees has been progressing sporadically since its inception by Sierpiński in 1933, the field has gained momentum beginning with the characterization of exact big Ramsey degrees of the Rado graph in [18] and more recently, with new techniques developed in [8] to handle triangle-free graphs. We refer to the paper [10] for comprehensive background on historical and recent results on big Ramsey degrees. This paper extends and concludes work in [8], [9], [11], [12], [13], [14], [17], [18], [25], [28], and [32] for free amalgamation classes with relations of arity at most two and finitely many forbidden irreducible substructures.…”