LARGE BOOK-CYCLE RAMSEY NUMBERS

摘要

Let B-n((k)) be the book graph which consists of n copies of Kk+1 all sharing a common Kk, and let Cm be a cycle of length m. In this paper, we first determine the exact value of r(B-n((2)) n, Cm) for 8 9 n + 112 leq m leq lceil 3n 2 ceil + 1 and n geq 1000. This answers a question of Faudree, Rousseau, and Sheehan [Ars Combin., 31 (1991), pp. 239--248] in a stronger form when m and n are large. Building upon this exact result, we are able to determine the asymptotic value of r(B(k) n, Cn) for each k geq 3. Namely, we prove that for each k geq 3, r(B (k) n, Cn) = (k + 1 + ok(1))n. This extends a result due to Rousseau and Sheehan [J. London Math. Soc., 18 (1978), pp. 392--396].