Publications of (and about) Paul Erdös
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 > s0(k), the set A = < nk: n = 1,2,... > has a partition A = \cupooj = 1Aj such that each Aj 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)].
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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