Some three-color Ramsey numbers, and

For given graphs G 1 , G 2 , G 3 , the three-color Ramsey number R ( G 1 , G 2 , G 3 ) is defined to be the least positive integer n such that every 3-coloring of the edges of complete graph K n contains a monochromatic copy of G i colored with i , for some 1 ≤ i ≤ 3 . In this paper, we prove that R ( P 4 , P 5 , C 3 ) = 11 , R ( P 4 , P 5 , C 4 ) = 7 , R ( P 4 , P 5 , C 5 ) = 11 , R ( P 4 , P 5 , C 7 ) = 11 , R ( P 4 , P 5 , C k ) = k + 2 for k ≥ 23 ; R ( P 4 , P 6 , C 4 ) = 8 , R ( P 4 , P 6 , C 3 ) = R ( P 4 , P 6 , C 5 ) = R ( P 4 , P 6 , C 7 ) = 13 , R ( P 4 , P 6 , C k ) = k + 3 for k ≥ 18 .