Determine Ramsey numbers for four or more colors

Determine upper and lower bounds for the Ramsey numbers R4(L) and Rk(L), where Rk(L) is the least grid side length such that every k-coloring of the integer grid contains a monochromatic L.

Background

The paper focuses mainly on the three-color Ramsey number R3(L), improving its upper bound while retaining the known lower-bound discussion. It notes that the corresponding Ramsey numbers for four colors and for general k remain unresolved and explicitly asks for both upper and lower bounds.

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