More results in polychromatic Ramsey theory

Abstract

Abstract. We study polychromatic Ramsey theory with a focus on colourings of [ω 2 ] 2 . We show that in the absence of GCH there is a wide range of possibilities. In particular each of the following is consistent relative to the consistency of ZFC:(1) 2 ω = ω 2 and ω 2 → poly (α) 2 ℵ 0 −bdd for every α < ω 2 . (2) 2 ω = ω 2 and ω 2 poly (ω 1 ) 2 2−bdd

Separating club-guessing principles in the presence of fat forcing axioms

“…Finally we present our main separation theorems, to be proved in Section 3. 1 Kunen's Axiom (KA), also known as Interval Hitting Principle (see for example [10]), is the following statement first considered by Kunen: There is a ladder system hC | 2 Lim(! 1 )i with the property that for every club C ✓ !…”

“…( [6]; cf. [21], [14], [1]) Let P ✓ H(), and let N be a collection of countable subsets of H(). We will say that N is a P -symmetric system i↵ the following holds.…”

“…Polychromatic Ramsey theory 8 (see [9], [2] and [1]) is the study of colourings f of sets of the form [X] n , focusing on the existence or non-existence of large sets Y ✓ X such that c [Y ] n is a one-toone function (such a set Y is usually called a rainbow for c). Here we consider the following negative polychromatic partition relation (see [1]).…”

“…Here we consider the following negative polychromatic partition relation (see [1]). Starting from CH and using symmetric systems, 9 Abraham and Cummings generically add a function c such that…”

“…The corresponding models are obtained as forcing extensions via constructions of either the first or the second type. Also, using a construction of the first type we answer a question of Abraham-Cummings in the context of polychromatic Ramsey theory ( [1]). …”