Open questions on improving bounds and constructing larger L-free colorings

Determine whether interval sequencing, properties of diagonals below the main diagonal, or properties of the subdiagonal of length n−1 can further improve the upper bound for the three-color Ramsey number R3(L); determine upper and lower bounds for the Ramsey numbers R4(L) and Rk(L); and construct, if possible, a 3-coloring of [22]×[22] with no monochromatic L, potentially using SAT solvers or artificial-intelligence and machine-learning techniques.

Background

The paper studies the Ramsey number R3(L), where L is the configuration consisting of three grid points forming an axis-aligned right angle with equal horizontal and diagonal offsets. The authors improve the previously known upper bound for R3(L) from 2593 to 493 using interval-counting arguments, diagonal properties, and Golomb rulers, while discussing computational attempts to improve the lower bound through SAT solving.

The concluding section identifies several unresolved directions: exploiting interval sequences between nonconsecutive red points, using additional diagonal structures, determining the corresponding Ramsey numbers for four or arbitrarily many colors, and searching computationally for a 22×22 three-coloring avoiding monochromatic L. These are presented collectively by the authors as open problems.