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

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