2025/01/04 by Dhruv Mubayi, Mubayi, Dhruv
Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2501.02389
openalex publication_date 2025/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fix integers d,r≥ 2 and suppose that the edge set of the d-fold Cartesian product of the N-clique KNd is r-colored. We show that there is a copy of Knd whose edges in each direction are monochromatic provided N > 2^2^c nd-1, where c depends only on r and d. This improves the previous best exponent of nd proved by Girão, Kronenberg, and Scott while also improving the best known bound due to them for a multidimensional Erd\H os-Szekeres monotone subsequence theorem introduced by Fishburn and Graham.