A partition theorem for a large dense linear order

Abstract

Let κ be a cardinal which is measurable after generically adding κ+ω many Cohen subsets to κ, and let Qκ = (Q, ≤ Q ) be the strongly κ-dense linear order of size κ. We prove, for 2 ≤ m < ω, that there is a finite value t + m such that the set [Q] m of m-tuples from Q can be partitioned into classes C i : i < t + m such that for any coloring a class C i in fewer thanthat is, for any coloring of [Qκ] m with fewer than κ colors there is a copy Q ⊆ Q of Qκ such that [Q ] m has at most t + m colors. On the other hand, we show that there are colorings of Qκ such that if Q ⊆ Q is any copy of Qκ then C i ∩ [Q ] = ∅ for all i < t + m , and hence Qκ [Qκ] m t + m .We characterize t + m as the cardinality of a certain finite set of ordered trees and obtain an upper and a lower bound on its value. In particular, t + 2 = 2 and for m > 2 we have t + m > tm, the m-th tangent number. The stated condition on κ is the hypothesis for a result of Shelah on which our work relies. A model in which this condition holds simultaneously for all m can be obtained by forcing from a model with a κ +ω -strong cardinal, but it follows from earlier results of Hajnal and Komjáth that our result, and hence Shelah's theorem, is not directly implied by any large cardinal assumption.

Partitions of large Rado graphs

“…In [1] we obtained a partition theorem for κ-dense linear orders whose statement is analogous to the statement of our Theorem 1.1. The relevant types are much more complicated for graphs than for orders and to capture them we need a better Diagonalization Lemma than the one provided in [1]; this is achieved in Lemma 3.3.…”

“…Partition properties of the random graph were studied by several authors, one can see a detailed discussion in J. Larson's paper [5]. Using a technique we introduced in our paper [1] along with a fine analysis of the types of finite subsets of G κ we are able to prove the following partition theorem. By G we mean a fixed copy of G κ whose universe is κ. Theorem 1.1.…”

“…Many historical points and a discussion of the use and to a certain extent necessity of the large cardinal assumptions we have are given in [1].…”

“…The relevant types are much more complicated for graphs than for orders and to capture them we need a better Diagonalization Lemma than the one provided in [1]; this is achieved in Lemma 3.3. Also we must work harder to show that the relevant types are realized by the ranges of the maps we use to build our copies of the Rado graph inside itself; this is achieved in Corollary 5.7.…”

“…Lemma 1.6. from [1] shows that being the range of a strong embedding is equivalent to being a strongly embedded subset which has another, somewhat more combinatorial definition.…”