Ramsey equivalence of $K_n$ and $K_n+K_{n-1}$

TL;DR

This paper proves that for all n ≥ 4, the complete graph K_n and the graph formed by adding a K_{n-1} to K_n are Ramsey equivalent, meaning they produce the same monochromatic cliques in edge colorings.

Contribution

It establishes the Ramsey equivalence between K_n and K_n+K_{n-1} for all n ≥ 4, a previously unknown relationship in graph Ramsey theory.

Findings

K_n and K_n+K_{n-1} are Ramsey equivalent for n ≥ 4

Any coloring producing a monochromatic K_n also produces a monochromatic K_n+K_{n-1}

The result extends understanding of Ramsey properties of complete graphs and their augmentations.

Abstract

We prove that, for $n\geq 4$, the graphs $K_n$ and $K_n+K_{n-1}$ are Ramsey equivalent. That is, if $G$ is such that any red-blue colouring of its edges creates a monochromatic $K_n$ then it must also possess a monochromatic $K_n+K_{n-1}$.