Find a 22-by-22 three-coloring without a monochromatic L

Construct a 3-coloring of the 22×22 integer grid [22]×[22] containing no monochromatic L, potentially using SAT solvers or artificial-intelligence/machine-learning techniques.

Background

The authors use SAT solvers to find a 20×20 three-colored grid with no monochromatic L and report that their attempts to resolve the 21×21 case had not produced a satisfiable or unsatisfiable result. They then speculate that a 22×22 example may exist and explicitly pose finding such a coloring as an open problem.

References

  1. Open Problems We close this paper with some open problems regarding the “L” problem. * Can interval sequencing (as detailed in Section 2.2 before Theorem 4) be used to further improve the upper bound?* Can properties of diagonals below the main diagonal and subdiagonal of lengthn − 1 be used to improve the upper bound?* What are upper and lower bounds for R4(L)? Rk(L)?* Though not found, we speculate that a 3-coloring of [22] × [22] with no monochromatic L exists. Try to find one, perhaps by using SAT solvers or AI/ML techniques.
— Ramsey Theory on the Integer Grid: The "L" Problem  (2502.05162 - Mammel et al., 7 Feb 2025) in Section 4, “Open Problems”