Partitions of large Rado graphs

Abstract

Let κ be a cardinal which is measurable after generically adding κ+ω many Cohen subsets to κ and let G = (κ, E) be the κ-Rado graph. We prove, for 2 ≤ m < ω, that there is a finite value r + m such that the set [κ] m can be partitioned into classes˙Ci : i < r + m¸s uch that for any coloring of any of the classes Ci in fewer than κ colors, there is a copy, that is, for any coloring of [G] m with fewer than κ colors there is a copy G of G such that [G ] m has at most r We characterize r + m as the cardinality of a certain finite set of types and obtain an upper and a lower bound on its value. In particular, r + 2 = 2 and for m > 2 we have r + m > rm where rm is the corresponding number of types for the countable Rado graph.

The Ramsey Theory of Henson graphs

“…This proof informed the approach in [5] to reduce the large cardinal assumption for obtaining the consistency of the Halpern-Läuchli Theorem at a measurable cardinal. Building on this and ideas from [36] and [6], Zhang proved the consistency of Laver's result for the κ-rationals, for κ measurable, in [40].…”

“…Theorem 2.4 (Halpern-Läuchli, [15]). Let T i = 2 <ω for each i < d, where d is any positive integer, and let (6) c :…”

“…We point out that Milliken's Theorem has been shown to consistently hold at a measurable cardinal by Shelah in [36], using ideas from Harrington's proof. An enriched version was proved by Džamonja, Larson, and Mitchell in [6] and applied to obtain the consistency of finite big Ramsey degrees for colorings of finite subsets of the κ-rationals, where κ is a measurable cardinal. They obtained the consistency of finite big Ramsey degrees for colorings of finite subgraphs of the κ-Rado graph for κ measurable in [6].…”

“…An enriched version was proved by Džamonja, Larson, and Mitchell in [6] and applied to obtain the consistency of finite big Ramsey degrees for colorings of finite subsets of the κ-rationals, where κ is a measurable cardinal. They obtained the consistency of finite big Ramsey degrees for colorings of finite subgraphs of the κ-Rado graph for κ measurable in [6]. The uncountable height of the tree 2 <κ coding the κ-rationals and the κ-Rado graph renders the notion of strong similarity type more complex than for the countable cases.…”

“…We say that two trees (T, ⊆) and (S, ⊆) are strongly similar trees if they satisfy Definition 3.1 in [35]. This is the the same as modifying Definition 4.3 by deleting (6) and changing (7) to apply to passing numbers of all nodes in the trees. By saying that two finite trees are strongly similar trees, we are implicitly assuming that their extensions to their immediate successors of their maximal nodes are still strongly similar.…”