On the Sumset Partition Problem

Abstract

Abstract A sequence m 1 , m 2 , … , m k of positive integers is n-realizable if there is a partition X 1 , X 2 , … , X k of the set [ n ] such that the sum of the elements in X i is m i for each i = 1 , 2 , … , k . In this paper we study the n-realizable sequences by adopting a different viewpoint from the one which has been previously used in the literature. We consider the realizability of a sequence in terms of the length k of the sequence or the values of its distinct elements. We characterize all n-realizable sequences with k ⩽ 4 and, for k ⩾ 5 , we prove that n ⩾ 4 k − 1 and m k > 4 k − 1 are sufficient conditions for a sequence to be n-realizable. This sufficient condition complements the one which has been used in connection with the ascending subgraph decomposition conjecture of Alavi et al. [Y. Alavi, A. J.Boals, G. Chartrand, P. Erdős and O. Oellerman. The ascending subgraph decomposition problem. Congressus Numeratium, 58 (1987), 7–14]. We also characterize realizable sequences whose distinct terms grow exponentially. Finally we consider the modular version of the problem and prove that all sequences in Z p of length k ⩽ ( p − 1 ) / 2 are realizable for any prime p ⩾ 3 . The bound on k is best possible.