Weak Schur numbers and the search for G.W. Walker’s lost partitions

Abstract

A set A of integers is weakly sum-free if it contains no three distinct elements x, y, z such that x + y = z. Given k ≥ 1, let WS(k) denote the largest integer n for which {1, . . . , n} admits a partition into k weakly sum-free subsets. In 1952, G.W. Walker claimed the value WS(5) = 196, without proof. Here we show WS(5) ≥ 196, by constructing a partition of {1, . . . , 196} of the required type. It remains as an open problem to prove the equality. With an analogous construction for k = 6, we obtain WS(6) ≥ 572. Our approach involves translating the construction problem into a Boolean satisfiability problem, which can then be handled by a SAT solver.