Bounds for Ramsey numbers of complete graphs dropping an edge

Let Kn−eKn−e be a graph obtained from a complete graph of order nn by dropping an edge, and let GpGp be a Paley graph of order pp. It is shown that if GpGp contains no Kn−eKn−e, then r(Kn+1−e)≥2p+1r(Kn+1−e)≥2p+1. For example, G1493G1493 contains no K13−eK13−e, so r(K14−e)≥2987r(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.