Partitions of bases into disjoint unions of bases (Q1103000)

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 \(A=<n^k:\) \(n=1,2,\ldots>\) has a partition \(A = \cup^{\infty}_{j=1}A_j\) 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 \textit{P. Erdős} and \textit{R. Rado} [Intersection theorems for system of sets, J. Lond. Math. Soc. 35, 85--90 (1960; Zbl 0103.27901)] and the reviewer [Homogeneous additive congruences, Philos. Trans. R. Soc. Lond., Ser. A 261, 163--210 (1967; Zbl 0139.27102)].