Exact values for some generalized Ramsey numbers

摘要

For graphs G_, G_2, …, G_r, the r-color Ramsey number R(G_1, G_2, …, G_r) is the least positive integer n such that every r-coloring of the edges of complete graph K_n contains a monochromatic copy of G_i colored with i, for some i ∈ {1, 2, …, r}. In this paper, by applying an integer linear programming model, we obtain that 39 new 2-color or 3-color Ramsey numbers in mixed cases. In particular, we determine the smallest open case n = 22 for R(C_4, K_(1,n)) and n = 12 for R(C_6, K_(1,n)).