The Gamma-Switch Ramsey Number

TL;DR

This paper introduces the $ extGamma$-switch Ramsey number, a new variation of classical Ramsey numbers involving group permutations of colours, and establishes bounds and exact values for specific cases.

Contribution

It defines the $ extGamma$-switch Ramsey number, develops initial theoretical bounds, and computes exact values for particular instances involving cyclic groups.

Findings

Proved $R_{C_3}(4,4,4) = R(3,3,3) + 1

Established $R_{C_4}(3,4,3,4) = R(2,3,2,3) + 1

Bounded $R_{C_4}(4,4,4,4)$ between $43$ and $R(3,3,3,3) + 1$

Abstract

We define and develop preliminary theoretical results for the $\Gamma$-switch Ramsey number, a variation on the classical $m$-colour Ramsey number for which we allow permuting the colours incident with a vertex using elements of a group $\Gamma \leq S_m$. We find bounds for the $\Gamma$-switch Ramsey number for groups with various properties as a function of the classical parameter. We prove $R_{C_3}(4,4,4) = R(3,3,3) + 1$, $R_{C_4}(3,4,3,4) = R(2,3,2,3) + 1$ and $43 \leq R_{C_4}(4,4,4,4) \leq R(3,3,3,3) + 1$.