On minimal Ramsey graphs and Ramsey equivalence in multiple colours

Abstract

For an integer $q\ge 2$, a graph $G$ is called $q$-Ramsey for a graph $H$ if every $q$-colouring of the edges of $G$ contains a monochromatic copy of $H$. If $G$ is $q$-Ramsey for $H$, yet no proper subgraph of $G$ has this property then $G$ is called $q$-Ramsey-minimal for $H$. Generalising a statement by Burr, Ne\v{s}et\v{r}il and R\"odl from 1977 we prove that, for $q\ge 3$, if $G$ is a graph that is not $q$-Ramsey for some graph $H$ then $G$ is contained as an induced subgraph in an infinite number of $q$-Ramsey-minimal graphs for $H$, as long as $H$ is $3$-connected or isomorphic to the triangle. For such $H$, the following are some consequences. (1) For $2\le r< q$, every $r$-Ramsey-minimal graph for $H$ is contained as an induced subgraph in an infinite number of $q$-Ramsey-minimal graphs for $H$. (2) For every $q\ge 3$, there are $q$-Ramsey-minimal graphs for $H$ of arbitrarily large maximum degree, genus, and chromatic number. (3) The collection $\{{\cal M}_q(H) : H \text{ is 3-connected or } K_3\}$ forms an antichain with respect to the subset relation, where ${\cal M}_q(H)$ denotes the set of all graphs that are $q$-Ramsey-minimal for $H$. We also address the question which pairs of graphs satisfy ${\cal M}_q(H_1)={\cal M}_q(H_2)$, in which case $H_1$ and $H_2$ are called $q$-equivalent. We show that two graphs $H_1$ and $H_2$ are $q$-equivalent for even $q$ if they are $2$-equivalent, and that in general $q$-equivalence for some $q\ge 3$ does not necessarily imply $2$-equivalence. Finally we indicate that for connected graphs this implication may hold: Results by Ne\v{s}et\v{r}il and R\"odl and by Fox, Grinshpun, Liebenau, Person and Szab\'o imply that the complete graph is not $2$-equivalent to any other connected graph. We prove that this is the case for an arbitrary number of colours.