Chromatic number is Ramsey distinguishing

A graph \(G\) is Ramsey for a graph \(H\) if every colouring of the edges of \(G\) in two colours contains a monochromatic copy of \(H\). Two graphs \(H_1\) and \(H_2\) are Ramsey equivalent if any graph \(G\) is Ramsey for \(H_1\) if and only if it is Ramsey for \(H_2\). A graph parameter \(s\) is Ramsey distinguishing if \(s(H_1)\neq s(H_2)\) implies that \(H_1\) and \(H_2\) are not Ramsey equivalent. In this paper we show that the chromatic number is a Ramsey distinguishing parameter. We also extend this to the multi-colour case and use a similar idea to find another graph parameter which is Ramsey distinguishing.