Bounds for Ramsey numbers of complete graphs dropping an edge

Let K n − e be a graph obtained from a complete graph of order n by dropping an edge, and let G p be a Paley graph of order p . It is shown that if G p contains no K n − e , then r ( K n + 1 − e ) ≥ 2 p + 1 . For example, G 1493 contains no K 13 − e , so r ( K 14 − e ) ≥ 2987 , improving the old bound 2557. It is also shown that r ( K 2 ¯ + G ) ≤ 4 r ( G , K 2 ¯ + G ) − 2 , implying that r ( K n − e ) ≤ 4 r ( K n − 2 , K n − e ) − 2 .