On some three color Ramsey numbers for paths and cycles

Abstract

For given graphs G 1 , G 2 , … , G k , k ≥ 2 , the multicolor Ramsey number, denoted by R ( G 1 , G 2 , … , G k ) , is the smallest integer n such that if we arbitrarily color the edges of a complete graph on n vertices with k colors, there is always a monochromatic copy of G i colored with i , for some 1 ≤ i ≤ k . Let P k (resp. C k ) be the path (resp. cycle) on k vertices. In the paper we consider the value for numbers of type R ( P i , P k , C m ) for odd m , k ≥ m ≥ 3 and k > 3 i 2 − 14 i + 25 4 when i is odd, and k > 3 i 2 − 10 i + 20 8 when i is even. In addition, we provide the exact values for Ramsey numbers R ( P 3 , P k , C 4 ) for all integers k ≥ 3 .