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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.
05C55 Generalized Ramsey theory
05C38 Paths and cycles
05C35 Extremal problems in graph theory
Full Text: DOI
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