The Ramsey Theory of Henson graphs

Abstract

For k ≥ 3, the Henson graph H k is the analogue of the Rado graph in which k-cliques are forbidden. Building on the author's result for H 3 in [4], we prove that for each k ≥ 4, H k has finite big Ramsey degrees: To each finite k-clique-free graph G, there corresponds an integer T (G, H k ) such that for any coloring of the copies of G in H k into finitely many colors, there is a subgraph of H k , again isomorphic to H k , in which the coloring takes no more than T (G, H k ) colors.Prior to this article, the Ramsey theory of H k for k ≥ 4 had only been resolved for vertex colorings by El-Zahar and Sauer in [7]. We develop a unified framework for coding copies of H k into a new class of trees, called strong H kcoding trees, and prove Ramsey theorems for these trees, forming a family of Halpern-Läuchli and Milliken-style theorems which are applied to deduce finite big Ramsey degrees. The approach here streamlines the one in [4] for H 3 and provides a general methodology opening further study of big Ramsey degrees for homogeneous structures with forbidden configurations. The results have bearing on topological dynamics via work of Kechris, Pestov, and Todorcevic [18] and Zucker [41].

Big Ramsey degrees using parameter spaces

“…The main pigeonhole argument is a technically challenging structured tree theorem, where the tree is built using a particular enumeration of the graph Henson H in which certain tree levels are coding (and contain vertices of the graph being represented) while others are branching. This method was later generalized to (non-oriented) Henson graphs [Dob19]. Recently, Zucker simplified it and further generalized to finitely constrained free amalgamation classes of structures in binary languages [Zuc20].…”

Colouring homogeneous structures

“…Which recently has seen two remarkable results. The solution of the very difficult open problem dealing with partitions of the set of finite substructures of the K n -free homogeneous graphs by Natasha Dobrinen, see [11]. A considerably more intricate situation than dealt with in this article, in which only partitions of the set of elements are investigated.…”

“…The article, see [12], by Jan Hubička and Jaroslav Nešetřil constituting, after a long history, in some sense a complete solution concerning Ramsey classes. The introduction of [11], provides an excellent exposition on the background and history of the partition theory of homogeneous structures. The work on partitions of sets of other structures than single elements has a long history.…”

Big Ramsey degrees of 3-uniform hypergraphs

“…One of the difficulties is the fact that Ramsey degrees are often determined by the number of non-isomorphic orderings (thus being 1 for classes of structures with ordering on vertices), while big Ramsey degrees are surprisingly rich (as discussed later). The interest in the area was recently renewed by the work of Zucker [8], which gives a good equivalent of Ramsey expansion for big Ramsey degrees, and by deep results of Dobrinen [3,4] about big Ramsey degrees of Henson graphs (see [4]).…”