Bounds for Ramsey numbers of complete graphs dropping an edge

Abstract

Let Kn−e be a graph obtained from a complete graph of order n by dropping an edge, and let Gp be a Paley graph of order p. It is shown that if Gp contains no Kn−e, then r(Kn+1−e)≥2p+1. For example, G1493 contains no K13−e, so r(K14−e)≥2987, improving the old bound 2557. It is also shown that r(K2¯+G)≤4r(G,K2¯+G)−2, implying that r(Kn−e)≤4r(Kn−2,Kn−e)−2.