Deleting Edges from Ramsey Minimal Examples

TL;DR

This paper proves that removing any edge from a Ramsey minimal example graph Kp destroys its Ramsey property, meaning the graph can then be edge-colored to avoid monochromatic Ks and Kt subgraphs.

Contribution

It establishes that deleting an edge from a Ramsey minimal example results in a non-Ramsey graph, confirming a key property of these minimal structures.

Findings

Removing an edge from Kp destroys its Ramsey property

The paper provides an accessible explanation of the proof process

It supports the conjecture about the structure of Ramsey minimal examples

Abstract

If r(s, t) = p, we shall call Kp a Ramsey Minimal Example. We prove that if r{s,t) = p, and one edge is deleted from the Ramsey Minimal Example, Kp, the resulting graph no longer has the Ramsey property, that is, its edges can be colored red or blue so that the "red" subgraph does not contain Ks and the "blue" subgraph does not contain Kt. The paper is written in an accessible style explaining the process that led us to make the conjecture and then prove this result.