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**Zentralblatt MATH**

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

**Zbl.No: ** 331.05122

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

**Title: ** The size Ramsey number. (In English)

**Source: ** Period. Math. Hung. 9, 145-161 (1978).

**Review: ** Let *C* be the class of all graphs G which satisfy G ––> (G_{1},G_{2}). As a way of measuring minimality for members of *C*, we define the size Ramsey number \hat r(G_{1},G_{2}) by \hat r(G_{1},G_{2}) = **max**_{G in C} |E(G)|. As usual, \hat r(G) signifies \hat r(G,G). For comparison purposes, we let \hat R(G_{1},G_{2}): = \binom{r(G_{1},G_{2})}{2}, where r(G_{1},G_{2}) denotes the standard Ramsey number. It is clear that \hat r(G_{1},G_{2}) \leq \hat R(G_{1},G_{2}) and we note in the paper that for all m,n, \hat r(K_{m},K_{n}) = \hat R(K_{m},K_{n}). On the other hand, \hat r can be much less than \hat R, a notion made precise by the following definition. An infinite sequence of graphs **{**G_{n}**}** is said to be an o-sequence if \hat r(G_{n}) = o(\hat R(G)n)) (n ––> oo). We prove several theorems related to the o-sequence concept. For example, we prove that if m is fixed, then **{**K_{m,n}**}** is an o-sequence. In the course of this work, we find some new results for standard Ramsey numbers. For example, letting K_{m} * \bar K_{n} denote the graph obtained from K_{m} by making one of its vertices adjacent to n additional vertices, we prove that if m is fixed and n is sufficiently large, then r(K_{m} * \bar K_{n}) = (m-1)(m+n-1)+1.

**Reviewer: ** P.Erdös

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

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