On SAT Solvers and Ramsey-type Numbers

We created and parallelized two SAT solvers to find new bounds on some Ramsey-type numbers. For \(c > 0\), let \(R_c(L)\) be the least \(n\) such that for all \(c\)-colorings of the \([n]\times [n]\) lattice grid there will exist a monochromatic right isosceles triangle forming an \(L\). Using a known proof that \(R_c(L)\) exists we obtained \(R_3(L) \leq 2593\). We formulate the \(R_c(L)\) problem as finding a satisfying assignment of a boolean formula. Our parallelized probabilistic SAT solver run on eight cores found a 3-coloring of \(20\times 20\) with no monochromatic \(L\), giving the new lower bound \(R_3(L) \geq 21\). We also searched for new computational bounds on two polynomial van der Waerden numbers, the "van der Square" number \(R_c(VS)\) and the "van der Cube" number \(R_c(VC)\). \(R_c(VS)\) is the least positive integer \(n\) such that for some \(c > 0\), for all \(c\)-colorings of \([n]\) there exist two integers of the same color that are a square apart. \(R_c(VC)\) is defined analogously with cubes. For \(c \leq 3\), \(R_c(VS)\) was previously known. Our parallelized deterministic SAT solver found \(R_4(VS)\) = 58. Our parallelized probabilistic SAT solver found \(R_5(VS) > 180\), \(R_6(VS) > 333\), and \(R_3(VC) > 521\). All of these results are new.