The Gamma-Switch Ramsey Number

We define and develop preliminary theoretical results for the $\mathrm{\Gamma }$-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 $\mathrm{\Gamma }\le S_m$. We find bounds for the $\mathrm{\Gamma }$-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 $43\le R_{C_4}(4,4,4,4)\le R(3,3,3,3)+1$.