Improved bounds for the minimum degree of minimal multicolor Ramsey graphs

Abstract

We provide two novel constructions of $r$ edge-disjoint $K_{k+1}$-free graphs on the same vertex set, each of which has the property that every small induced subgraph contains a complete graph on $k$ vertices. The main novelty of our argument is the combination of an algebraic and a probabilistic coloring scheme, which utilizes the beneficial algebraic and combinatorial properties of the Hermitian unital. These constructions improve on a number of upper bounds on the smallest possible minimum degree of minimal $r$-color Ramsey graphs for the clique $K_{k+1}$ when $r\ge c\frac{k}{{\log }^{2}k}$ and $k$ is large enough.