Sách A generalization of generalized Paley graphs and new lower bounds for R(3, q)

Abstract
Generalized Paley graphs are cyclic graphs constructed from quadratic or higher
residues of finite fields. Using this type of cyclic graphs to study the lower bounds
for classical Ramsey numbers, has high computing efficiency in both looking for
parameter sets and computing clique numbers. We have found a new generaliza-
tion of generalized Paley graphs, i.e. automorphism cyclic graphs, also having the
same advantages. In this paper we study the properties of the parameter sets of
automorphism cyclic graphs, and develop an algorithm to compute the order of the
maximum independent set, based on which we get new lower bounds for 8 classical
Ramsey numbers: R(3, 22) > 131, R(3, 23) > 137, R(3, 25) > 154, R(3, 28) > 173,
R(3, 29) > 184, R(3, 30) > 190, R(3, 31) > 199, R(3, 32) > 214. Furthermore, we
also get R(5, 23) > 521 based on R(3, 22) > 131. These nine results above improve
their corresponding best known lower bounds.