Minimal Ramsey Graphs with Many Vertices of Small Degree

Given any graph $H$, a $G$ is said to be $q$-Ramsey for $H$ if every coloring of the edges with $q$ colors yields monochromatic subgraph isomorphic $H$. Such minimal additionally no proper $G'$ In 1976, Burr, Erdös, and Lovász initiated study parameter $s_q(H)$, defined as smallest minimum degree among all graphs this paper, we consider problem determining how many vertices $s_q(H)$ can contain. Specifically, seek identify which contain arbitrarily such vertices. We call satisfying property $s_q$-abundant. Among other results, prove that cycle $s_q$-abundant integer $q\geq 2$. also discuss cases when clique or pendant edge, extending previous results Burr co-authors Fox co-authors. To our construct suitable Ramsey graphs, use gadget pattern gadgets generalize earlier constructions used in graphs. provide new, more constructive proof existence these gadgets.