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

TL;DR

This paper proves that the complete graph $K_n$ and the graph $K_n+K_{n-1}$ have identical $r$-Ramsey graphs for all $n, r \

Contribution

It extends previous results by providing a simple proof that these graphs are Ramsey equivalent for all numbers of colors $r \

Findings

$K_n$ and $K_n+K_{n-1}$ are $r$-Ramsey equivalent for all $n, r \\geq 3$

The proof generalizes earlier work from 2-color to multiple colors

The equivalence holds with no exceptions for $n \\geq 3$

Abstract

In 2015 Bloom and Liebenau proved that $K_n$ and $K_n+K_{n-1}$ possess the same $2$-Ramsey graphs for all $n\geq 3$ (with a single exception for $n=3$). In the following we give a simple proof that $K_n$ and $K_n+K_{n-1}$ possess the same $r$-Ramsey graphs for all $n, r\geq 3$.