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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 383.05027

**Autor: ** Erdös, Paul; Faudree, Ralph J.; Rousseau, C.C.; Schelp, R.H.

**Title: ** On cycle-complete graph Ramsey numbers. (In English)

**Source: ** J. Graph Theory 2, 53-64 (1978).

**Review: ** Given graphs G_{1} and G_{2}, there exists an integer r such that if q is any integer greater than or equal to r, if E_{1},E_{2} is any partition of the edge set of K_{q} (the complete graph on q vertices) and if H_{1} and H_{2} are the subgraphs of K_{q} with these edge sets, then either H_{1} contains a subgraph isomorphic to G_{1} or H_{2} contains a subgraph isomorphic to G_{2}. The Ramsey number r(G_{1},G_{2}) is the smallest integer with the obove property. In this paper the authors consider the case where G_{1} is C_{m}, a circuit of length m, and G_{2} is K_{n}. The main result is: for all m \geq 3 and n \geq 2, r(C_{m},K_{n}) \leq **{**(m-2)(n^{1/k}+2)+1**}**(n-1), where **{**x**}** denotes the least inter \geq x and k denotes the integer part of (m-1)/2. Additional results are given for special values of m or n.

**Reviewer: ** J.E.Graver

**Classif.: ** * 05C55 Generalized Ramsey theory

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