On the number of high‐dimensional partitions

Abstract

AbstractLet denote the number of ‐dimensional partitions with entries from . Building upon the works of Balogh–Treglown–Wagner and Noel–Scott–Sudakov, we show that when , holds for all . This makes progress toward a conjecture of Moshkovitz–Shapira [Adv. in Math. 262 (2014), 1107–1129]. Via the main result of Moshkovitz and Shapira, our estimate also determines asymptotically a Ramsey‐theoretic parameter related to Erdős–Szekeres‐type functions, thus solving a problem of Fox, Pach, Sudakov, and Suk [Proc. Lond. Math. Soc. 105 (2012), 953–982]. Our main result is a new supersaturation theorem for antichains in , which may be of independent interest.