Vertex Folkman Numbers and the Minimum Degree of Minimal Ramsey Graphs

We investigate the smallest possible minimum degree of $r$-color minimal Ramsey graphs for the $k$-clique. In particular, we obtain a bound of the form $O(k^2\log^2 k\big)$, which is tight up to a $(\log^2 k)$-factor whenever the number $r\geq2$ of colors is fixed. This extends the work of Burr, Erdos, and Lovasz, who determined this extremal value for two colors and any clique size, and complements that of Fox, Grinshpun, Liebenau, Person, and Szabo, who gave essentially tight bounds when the order $k$ of the clique is fixed. As a side product our result also yields an improved upper bound on the vertex Folkman number $F(r,k, k+1)$ of the $k$-clique. The proof relies on a reformulation of the corresponding extremal function by Fox et al. and combines and refines methods used by Dudek, Eaton, and Rodl.