On minimal Ramsey graphs and Ramsey equivalence in multiple colours
Dennis Clemens, Anita Liebenau, Damian Reding

TL;DR
This paper investigates the structure and properties of minimal Ramsey graphs across multiple colours, establishing new results on their abundance, diversity, and equivalence relations, especially for 3-connected graphs and triangles.
Contribution
It generalizes previous results by proving the existence of infinitely many minimal Ramsey graphs for certain graphs and explores the concept of Ramsey equivalence across different numbers of colours.
Findings
Infinite minimal Ramsey graphs for 3-connected or triangle graphs.
Existence of minimal Ramsey graphs with arbitrarily large degree, genus, and chromatic number.
The set of minimal Ramsey graphs forms an antichain under subset relation.
Abstract
For an integer , a graph is called -Ramsey for a graph if every -colouring of the edges of contains a monochromatic copy of . If is -Ramsey for , yet no proper subgraph of has this property then is called -Ramsey-minimal for . Generalising a statement by Burr, Ne\v{s}et\v{r}il and R\"odl from 1977 we prove that, for , if is a graph that is not -Ramsey for some graph then is contained as an induced subgraph in an infinite number of -Ramsey-minimal graphs for , as long as is -connected or isomorphic to the triangle. For such , the following are some consequences. (1) For , every -Ramsey-minimal graph for is contained as an induced subgraph in an infinite number of -Ramsey-minimal graphs for . (2) For every , there are -Ramsey-minimal graphs for of…
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