Determine bounds for four or more colors

Determine upper and lower bounds for R4(L) and, more generally, for Rk(L), where Rk(L) is the least positive integer n such that every k-coloring of the n × n integer grid contains a monochromatic L.

Background

The paper focuses primarily on the three-color parameter R3(L), improving its upper bound to 493 while retaining the known lower-bound framework. Although the introduction recalls the general Gallai–Witt guarantee for every number of colors, quantitative bounds beyond the three-color case are not established here. The authors explicitly identify the determination of bounds for R4(L) and Rk(L) as open problems.