Multicolor Ramsey Numbers Involving C4: Improved Bounds and an Equality

We establish new lower and upper bounds, as well as an equality, for multicolor Ramsey numbers involving the 4-cycle C4. An explicit four-coloring of K87 yields R(C4,C4,K4,K4) ≥ 88, improving the previous lower bound of 87. Together with a new upper bound, this gives 88 ≤ R(C4,C4,K4,K4) ≤ 173. We also prove that R(C4,K3+e,K4) = R(C4,K3,K4), where K3+e is a triangle with a pendant edge, and establish R(C4,K4−e,K4) ≤ 35 and R(C4,C4,K3,K3) ≤ 35, improving previous upper bounds of 36. Consequently, R(C4,C4,K3,K4) ≤ 73, improving 75. The upper bounds use structural and extremal arguments together with two finite SAT verifications. The new lower bound follows from an explicit coloring based on the Paley graph of order 17. The computational certificates, verification programs, and complete coloring matrix are available separately at https://doi.org/10.5281/zenodo.23959832.