GALLAI-RAMSEY NUMBERS FOR RAINBOW S3+ AND MONOCHROMATIC PATHS

Motivated by Ramsey theory and other rainbow-coloring-related problems, we consider edge-colorings of complete graphs without rainbow copy of some fixed subgraphs. Given two graphs G and H, the k-colored Gallai-Ramsey number gr(k) (G : H) is defined to be the minimum positive integer n such that every k-coloring of the complete graph on n vertices contains either a rainbow copy of G or a monochromatic copy of H. Let S-3(+) be the graph on four vertices consisting of a triangle with a pendant edge. In this paper, we prove that gr(k)(S-3(+) : P-5) = k + 4 (k >= 5), gr(k) (S-3(+) : mP(2)) = (m -1)k + m + 1 (k >= 1), gr(k) (S-3(+) : P-3 boolean OR P-2) = k + 4 (k >= 5) and gr(k) (S-3(+) : 2P(3)) = k + 5 (k >= 1).