Lower bounds for and from Paley graph and generalization

Let q ≡ 1 ( mod 4 ) be a prime power and P q the Paley graph of order q . It is shown that if P q contains no copy of G , where δ ( G ) ≥ 1 , then r 2 ( K 1 + G ) ≥ 2 q + 1 . In particular, if 4 n + 1 is a prime power, then r 2 ( K 3 + K ¯ n ) ≥ 8 n + 3 . Furthermore, the Paley graph P q for q = 1 ( mod 6 ) is generalized to H 0 ( q ) , H 1 ( q ) and H 2 ( q ) , which are ( q − 1 ) / 3 -regular, isomorphic to each other and form an edge-coloring of K q . It is shown that if H 0 ( q ) contains no copy of G with δ ( G ) ≥ 1 , then r 3 ( K 1 + G ) ≥ 3 q + 1 . Also, each pair of adjacent vertices in H 0 ( q ) has the same number of common neighbors. We shall compute this number for many H 0 ( p ) , where p is a prime for convenience of the algorithm. Each of computing data gives lower bounds for some three-color Ramsey numbers.