Lower bounds for [formula omitted] and [formula omitted] from Paley graph and generalization

Abstract

Let q≡1(mod4) be a prime power and Pq the Paley graph of order q. It is shown that if Pq contains no copy of G, where δ(G)≥1, then r2(K1+G)≥2q+1. In particular, if 4n+1 is a prime power, then r2(K3+K¯n)≥8n+3. Furthermore, the Paley graph Pq for q=1(mod6) is generalized to H0(q), H1(q) and H2(q), which are (q−1)/3-regular, isomorphic to each other and form an edge-coloring of Kq. It is shown that if H0(q) contains no copy of G with δ(G)≥1, then r3(K1+G)≥3q+1. Also, each pair of adjacent vertices in H0(q) has the same number of common neighbors. We shall compute this number for many H0(p), where p is a prime for convenience of the algorithm. Each of computing data gives lower bounds for some three-color Ramsey numbers.