Some partitions of positive integers

Abstract

Research problem 231, Discrete Mathematics 140 (1995) says: Let A be a set of 2k, k ≥ 2, distinct positive integers. It is desired to partition A into two subsets A0 and A1 each with cardinality k so that the sum of any k - 1 elements of Ai is not an element of Ai+1, i = 0, 1 mod 2. It is not possible to find such a partition when A is {1,3,4,5,6,7} or any of {1,2,3,4,5,x}, x ≥ 7. Can it be done in all other cases? We show that the answer is affirmative for k ≥ 3 with some exceptions for k = 3. © 1999 Elsevier Science B.V. All rights reserved.