A note on multidimensional Ramsey numbers
Dhruv Mubayi

TL;DR
This paper improves bounds on multidimensional Ramsey numbers, showing that large enough Cartesian products contain monochromatic subcubes, advancing understanding in high-dimensional combinatorics and Ramsey theory.
Contribution
It provides a significantly improved exponential bound for the existence of monochromatic subcubes in multidimensional Ramsey problems.
Findings
Established a new bound $N > 2^{2^{c n^{d-1}}}$ for monochromatic $K_n^d$
Improved previous bounds by Girão, Kronenberg, and Scott
Enhanced bounds for a multidimensional Erdős–Szekeres theorem
Abstract
Fix integers and suppose that the edge set of the -fold Cartesian product of the -clique is -colored. We show that there is a copy of whose edges in each direction are monochromatic provided , where depends only on and . This improves the previous best exponent of proved by Gir\~ao, 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.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Limits and Structures in Graph Theory
