Bounds on some Ramsey numbers involving quadrilateral

Xiaodong Xu
Zehui Shao
Stanislaw Radziszowski

Note: imported from RIT’s Digital Media Library running on DSpace to RIT Scholar Works in February 2013.


For graphs G_1,G_2,...,G_m the Ramsey number R(G_1,G_2,...,G_m) is defined to be the smallest integer n such that any m-coloring of the edges of the complete graph K_n must include a monochromatic G_i in color i, for some i. In this note we establish several lower and upper bounds for some Ramsey numbers involving quadrilateral C_4, including R(C_4,K_9) <= 32, 19 <= R(C_4,C_4,K_4) <= 22, 31 <= R(C_4,C_4,C_4,K_4) <= 50, 52 <= R(C_4,K_4,K_4) <= 72, 42 <= R(C_4,C_4,K_3,K_4) <= 76, and 87 <= (C_4,C_4,K_4,K_4) <= 179.