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**Zentralblatt MATH**

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

**Zbl.No: ** 645.10045

**Autor: ** Erdös, Paul; Nathanson, Melvyn B.

**Title: ** Partitions of bases into disjoint unions of bases. (In English)

**Source: ** J. Number Theory 29, No.1, 1-9 (1988).

**Review: ** Two Ramsay-like combinatorial results on partitions are proved using probabilistic methods and the Borel-Cantelli lemma. The authors deduce that if A is an asymptotic basis of order h and if every large integer has sufficiently many representations as a sum of h elements of A, then A is a union of a finite or infinite number of pairwise disjoint asymptotic bases of order h.

Waring's problem is extended to showing that for each k \geq 2 and for all s > s_{0}(k), the set A = < n^{k}: n = 1,2,... > has a partition A = \cup^{oo}_{j = 1}A_{j} such that each A_{j} is an asymptotic basic of order s. In the other direction, they show that the squares cannot be partitioned into disjoint sets which are asymptotic bases of order 4; for numbers not divisible by 4 there is a positive result. Some open problems are also included. For another combinatorial result which also has applications to additive number theory, see *P. Erdös* and *R. Rado* [Intersection theorems for system of sets, J. Lond. Math. Soc. 35, 85-90 (1960; Zbl 103.27901)] and the reviewer [Homogeneous additive congruences, Philos. Trans. R. Soc. Lond., Ser. A 261, 163-210 (1967; Zbl 139.27102)].

**Reviewer: ** M.M.Dodson

**Classif.: ** * 11B13 Additive bases

11B75 Combinatorial number theory

11P05 Waring's problem and variants

05C55 Generalized Ramsey theory

05A05 Combinatorial choice problems

11P81 Elementary theory of partitions

**Keywords: ** asymptotic basis of order h; Waring's problem; partition

**Citations: ** Zbl 103.27901; Zbl 139.27102

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