Bounds on Some Ramsey Numbers
نویسندگان
چکیده
For graphs G1, G2, · · · , Gm, the Ramsey number R(G1, G2, · · · , Gm) is defined to be the smallest integer n such that anym-coloring of the edges of the complete graphKn must include a monochromatic Gi in color i, for some i. In this note we establish 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. Supported by the National Natural Science Foundation of China (Grant Nos. 60533010, 60703047 and 60563008), the Basic Research Fund of Guangxi Academy of Sciences (080414), the Program for New Century Excellent Talents in University (NCET05-0612) and the Chenguang Program of Wuhan (200750731262).
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