Use additional diagonals to improve the upper bound

Investigate whether properties of the diagonals below the main diagonal and of the subdiagonal of length n−1 can be used to improve the upper bound for the three-color Ramsey number R3(L) of the monochromatic L configuration.

Background

The upper-bound arguments in the paper primarily exploit the main diagonal and subdiagonals below it. The authors identify further structural information in the diagonals below the main diagonal and in the subdiagonal of length n−1 as a possible source of stronger constraints, but leave unresolved whether those properties can improve the bound R3(L)≤493.

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?
— Ramsey Theory on the Integer Grid: The "L" Problem  (2502.05162 - Mammel et al., 7 Feb 2025) in Section 4, “Open Problems”