On the number of high-dimensional partitions

Let P-d(n) denote the number of n x center dot center dot center dot x n d-dimensional partitions with entries from {0,1, center dot center dot center dot, n}. Building upon the works of Balogh-Treglown-Wagner and Noel-Scott-Sudakov, we show that when d -> infinity, P-d(n)=2((1+od(1))) root 6/(d+1)pi-n(d) holds for all n >= 1. This makes progress towards 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 Erdos-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 [n](d), which may be of independent interest.