Improve the upper bound through interval sequencing

Determine whether sequencing intervals between nonconsecutive red points, as described in Section 2.2, can further improve the upper bound for the three-color Ramsey number R3(L) of the monochromatic L configuration on an integer grid.

Background

The paper improves the previously known upper bound for R3(L) from 2593 to 493 by analyzing intervals between red points on the main diagonal and by using Golomb rulers. Section 2.2 extends the interval method from consecutive red points to intervals between arbitrary pairs of red points, but the authors do not fully develop the resulting sequencing problem. They explicitly ask whether this approach can yield a further improvement.

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?
— Ramsey Theory on the Integer Grid: The "L" Problem  (2502.05162 - Mammel et al., 7 Feb 2025) in Section 4, “Open Problems,” following Section 2.2 and Theorem 4

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?

— Ramsey Theory on the Integer Grid: The "L" Problem  (2502.05162 - Mammel et al., 7 Feb 2025) in Section 4, page 19; interval sequencing is detailed in Section 2.2 before Theorem 4

Can interval sequencing (as detailed in Section 2.2 before Theorem 4) be used to further improve the upper bound?

— Ramsey Theory on the Integer Grid: The "L" Problem  (2502.05162 - Mammel et al., 7 Feb 2025) in Section 4, p. 19; interval-sequencing discussion in Section 2.2