Indestructible colourings and rainbow Ramsey theorems

TL;DR

This paper explores the consistency of certain negative partition relations and rainbow Ramsey theorems under various set-theoretic assumptions, demonstrating the independence of these combinatorial properties from ZFC.

Contribution

It provides new consistency results showing that specific rainbow Ramsey properties can fail even under GCH and Martin's Axiom, extending understanding of combinatorial set theory.

Findings

Existence of colorings with no embedded rainbow subcolorings under GCH.

Consistency of large continuum with negative rainbow partition relations.

Failure of certain rainbow Ramsey properties under Martin's Axiom.

Abstract

We give a negative answer to a question of Erdos and Hajnal: it is consistent that GCH holds and there is a colouring $c:[{\omega_2}]^2\to 2$ establishing $\omega_2 \not\to [(\omega_1;{\omega})]^2_2$ such that some colouring $g:[\omega_1]^2\to 2$ can not be embedded into $c$. It is also consistent that $2^{\omega_1}$ is arbitrarily large, and a function $g$ establishes $2^{\omega_1} \not\to [(\omega_1,\omega_2)]^2_{\omega_1}$ such that there is no uncountable $g$-rainbow subset of $2^{\omega_1}$. We also show that for each $k\in {\omega}$ it is consistent with Martin's Axiom that the negative partition relation $\omega_1 \not\to^* [(\omega_1;\omega_1)]_{k-bdd}$ holds.