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**Zentralblatt MATH**

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

**Zbl.No: ** 822.05046

**Autor: ** Bialostocki, Arie; Erdös, Paul; Lefmann, Hanno

**Title: ** Monochromatic and zero-sum sets of nondecreasing diameter. (In English)

**Source: ** Discrete Math. 137, No.1-3, 19-34 (1995).

**Review: ** The authors consider a van der Waerden type number f(m, r), which is the minimum integer n such that for every coloring of the integers **{**1, 2,..., n**}** with r colors, there exist two monochromatic subsets B_{1} and B_{2} each with m integers such that each element of B_{1} is less than each element of B_{2} and that the diameter of B_{1} is less than or equal to the diameter of B_{2}. They verify that f(m, 2) = 5m- 3, f(m, 3) = 9m- 7, 12m- 9 \leq f(m, 4) \leq 13m- 11, and asymptotically, c_{1} mr \leq f(m, r) \leq c_{2} mr log_{2} r. Similar questions are considered when the elements of **Z**_{m} are used as colors and zero-sum sets are required.

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

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

**Keywords: ** van der Waerden number; coloring; monochromatic subsets; diameter; zero- sum sets

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