A note on highly connected and well-connected Ramsey theory

TL;DR

This paper explores weakened forms of classical partition relations in graph theory, focusing on highly connected monochromatic subgraphs, and examines their consistency under various set-theoretic assumptions.

Contribution

It introduces and analyzes new weakenings of classical partition relations, showing their consistency at accessible cardinals and under different set-theoretic principles.

Findings

Weakened relations can hold at accessible cardinals where classical ones fail.

Large cardinals and forcing axioms influence these partition relations.

A non-trivial instance can hold at the continuum.

Abstract

We study a pair of weakenings of the classical partition relation $\nu \rightarrow (\mu)^2_\lambda$ recently introduced by Bergfalk-Hru\v{s}\'{a}k-Shelah and Bergfalk, respectively. Given an edge-coloring of the complete graph on $\nu$-many vertices, these weakenings assert the existence of monochromatic subgraphs exhibiting high degrees of connectedness rather than the existence of complete monochromatic subgraphs asserted by the classical relations. As a result, versions of these weakenings can consistently hold at accessible cardinals where their classical analogues would necessarily fail. We prove some complementary positive and negative results indicating the effect of large cardinals, forcing axioms, and square principles on these partition relations. We also prove a consistency result indicating that a non-trivial instance of the stronger of these two partition relations can hold at the continuum.