# Ramsey theory for monochromatically well-connected subsets

## TL;DR

This paper introduces a new notion of well-connectedness in infinite combinatorics, compares it with classical Ramsey relations, and explores its consistency and strength in various set-theoretic models.

## Contribution

It defines well-connectedness, analyzes its relation to classical Ramsey relations, and establishes its consistency and equiconsistency with large cardinal assumptions.

## Key findings

- Well-connectedness weakens classical Ramsey relations.
- In certain models, well-connectedness relations are consistent with classical ones.
- The strength of well-connectedness at _2 is equiconsistent with a weakly compact cardinal.

## Abstract

We define well-connectedness, an order-theoretic notion of largeness whose associated partition relations $\nu\to_{wc}(\mu)_\lambda^2$ formally weaken those of the classical Ramsey relations $\nu\to(\mu)_\lambda^2$. We show that it is consistent that the arrows $\to_{wc}$ and $\to$ are, in infinite contexts, essentially indistinguishable. We then show, in contrast, that in Mitchell's model of the tree property at $\omega_2$, the relation $\omega_2\to_{wc}(\omega_2)_\omega^2$ does hold, and that the consistency strength of this relation holding is precisely a weakly compact cardinal. These investigations may be viewed as augmenting those of [BHS], the central arrow of which, $\to_{hc}$, is of intermediate strength between $\to_{wc}$ and the Ramsey arrow $\to$.