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**Zentralblatt MATH**

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

**Zbl.No: ** 858.05073

**Autor: ** Erdös, Paul; Tuza, Zsolt; Valtr, Pavel

**Title: ** Ramsey-remainder. (In English)

**Source: ** Eur. J. Comb. 17, No.6, 519-532 (1996).

**Review: ** The following general question is considered: Given a positive integer k, find the minimum number rr(k) such that any sufficiently large set S belonging to some class *S* can be decomposed into ``regular'' sets of size at least k with a remainder set of size at most rr(k). The number rr(k) is the Ramsey-remainder. It is shown for example, that if *S* is the set of all posets, and regularity refers to a poset being a chain or an antichain, then rr(k) = (k-1)(k-2) = r(k,k-1), where r(k,k-1) is the poset Ramsey number. A similar result is proved when *S* is the class of all finite r-uniform complete hypergraphs the edges of which are colored by c colors and a regular hypergraph is one that is monochromatic. In this case rr(k) = r_{c,k}(k)-1. Other interesting Ramsey-remainder results are investigated, in particular, when *S* is the class of finite sets of points in general position in the plane and regularity refers to convexity. In this later case a sharp bound for the corresponding Ramsey-remainder number is obtained if the Erdös-Szekeres conjecture on the Ramsey number for convex sets in the plane is true.

**Reviewer: ** R.Faudree (Memphis)

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

05D10 Ramsey theory

**Keywords: ** Ramsey-remainder; Ramsey number; hypergraph; Erdös-Szekeres conjecture

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