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.

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