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**Zentralblatt MATH**

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

**Zbl.No: ** 329.05116

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

**Title: ** Generalized Ramsey theory for multiple colors. (In English)

**Source: ** J. Comb. Theory, Ser. B 20, 250-264 (1976).

**Review: ** From the authors' abstract: In this paper, we study the generalized Ramsey number r(G_{1}, ... ,G_{k}) where the graphs G_{1}, ... ,G_{k} consist of complete graphs, complete bipartite graphs, paths, and cycles. Our main theorem gives the Ramsey number for the case where G_{2}, ... ,G_{k} are fixed and G_{1} \cong C_{n} or P_{n} with n sufficiently large. If among G_{2}, ... ,G_{k} there are both complete graphs and odd cycles, the main theorem requires an additional hypothesis concerning the size of the odd cycles relative to their number. If among G_{2}, ... ,G_{k} there are odd cycles but no complete graphs, then no additional hypothesis is necessary and complete results can be expressed in terms of a new type of Ramsey number which is introduced in this paper. For k = 3 and k = 4 we determine all necessary values of the new Ramsey number and so obtain, in particular, explicit and complete results for the cycle Ramsey numbers r(C_{n},C_{l},C_{k}) and r(C_{n},C_{l},C_{k},C_{m}) when n is large.

**Reviewer: ** J.E.Graver

**Classif.: ** * 05C35 Extremal problems (graph theory)

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