Bounds on Some Ramsey Numbers Involving Quadrilateral

Xiaodong Xu, Guangxi Academy of Sciences, Nanning, China Zehui Shao, Huazhong University of Science and Technology, Wuhan, China Stanis law Radziszowski∗, Rochester Institute of Technology, NY, USA For graphs G1, G2, · · · , Gm, the Ramsey number R(G1, G2, · · · , Gm) is defined to be the smallest integer n such that any m-coloring of the edges of the complete graph Kn must include a monochromatic Gi in color i, for some i. In this talk we report on several lower and upper bounds for some Ramsey numbers involving quadrilateral C4, including R(C4, K9) ≤ 32, 19 ≤ R(C4, C4, K4) ≤ 22, 31 ≤ R(C4, C4, C4, K4) ≤ 50, 52 ≤ R(C4, K4, K4) ≤ 72, 42 ≤ R(C4, C4, K3, K4) ≤ 76, and 87 ≤ R(C4, C4, K4, K4) ≤ 179.