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The Ramsey number for a cycle of length six versus a clique of order eight. (English) Zbl 1159.05033
It was conjectured by P. Erdős, R.J. Faudree, C.C. Rousseau, and R.H. Schelp in [J. Graph Theory 2 , 53-64 (1978; Zbl 0383.05027)] that \(r(K_m, C_n) = (m - 1)(n - 1) + 1\) for \(3 \leq m \leq n\) and \((m,n) \neq (3,3)\), and the question was posed to find the smallest \(n\) for which this is true. It is shown that the Ramsey number \(r(K_8, C_6) = 36 = (8 - 1)(6 - 1) + 1\). The conjecture has been verified for \(m \leq 7\), and this adds to the results that support the conjecture of Erdős et.al.
MSC:
05C55 Generalized Ramsey theory
05C38 Paths and cycles
05C35 Extremal problems in graph theory
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