Weak Schur numbers and the search for G.W. Walkers lost partitions
A set A A of integers is weakly sum-free if it contains no three distinct  elements x,y,z x , y , z such that x+y=z x + y = z . Given k?1 k ? 1 , let WS(k) W S ( k ) denote the largest integer n n for which {1,,n} { 1 ,  , n } admits a partition into k k weakly sum-free subsets. In 1952, G.W.Walker claimed the value WS(5)=196 W S ( 5 ) = 196 , without proof. Here we show WS(5)?196 W S ( 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=6 k = 6 , we obtain WS(6)?572 W S ( 6 ) ? 572 . Our approach involves translating the construction problem into a Boolean satisfiability problem, which can then be handled by a SAT solver.
schur numbers sat solver sum-free sets weakly sum-free sets boolean variables sat problem