The value of the Ramsey number R4(C4)

Abstract

The multicolor Ramsey number R_r(H) is defined to be the smallest integer n = n(r) with the property that any r-coloring of the edges of the complete graph K_n must result in a monochromatic subgraph of K _n isomorphic to H. It is well known that 18 ≤ R₄[C ₄) ≤ 19. In this paper, with the help of a substantial amount of computation, we prove that R₄(C₄) = 18.