# Ramsey theory for highly connected monochromatic subgraphs

## TL;DR

This paper explores variants of Ramsey's Theorem focusing on the existence of large, highly connected monochromatic subgraphs within edge-colored complete graphs, emphasizing weaker connectivity conditions.

## Contribution

It introduces and analyzes weaker forms of Ramsey's Theorem related to highly connected monochromatic subgraphs in edge-colored complete graphs.

## Key findings

- Existence of large highly connected monochromatic subgraphs under weaker conditions
- Characterization of connectivity properties in colored complete graphs
- New theoretical insights into Ramsey-type problems for highly connected subgraphs

## Abstract

An infinite graph is highly connected if the complement of any subgraph of smaller size is connected. We consider weaker versions of Ramsey's Theorem asserting that in any coloring of the edges of a complete graph there exist large highly connected subgraphs all of whose edges are colored by the same color.