Dedekind's problem in the hypergrid

Abstract

Consider the partially ordered set on $[t]^n:=\{0,\dots,t-1\}^n$ equipped with the natural coordinate-wise ordering. Let $A(t,n)$ denote the number of antichains of this poset. The quantity $A(t,n)$ has a number of combinatorial interpretations: it is precisely the number of $(n-1)$-dimensional partitions with entries from $\{0,\dots,t\}$, and by a result of Moshkovitz and Shapira, $A(t,n)+1$ is equal to the $n$-color Ramsey number of monotone paths of length $t$ in 3-uniform hypergraphs. This has led to significant interest in the growth rate of $A(t,n)$. A number of results in the literature show that $\log_2 A(t,n)=(1+o(1))\cdot \alpha(t,n)$, where $\alpha(t,n)$ is the width of $[t]^n$, and the $o(1)$ term goes to $0$ for $t$ fixed and $n$ tending to infinity. In the present paper, we prove the first bound that is close to optimal in the case where $t$ is arbitrarily large compared to $n$, as well as improve all previous results for sufficiently large $n$. In particular, we prove that there is an absolute constant $c$ such that for every $t,n\geq 2$, $$\log_2 A(t,n)\leq \left(1+c\cdot \frac{(\log n)^3}{n}\right)\cdot \alpha(t,n).$$ This resolves a conjecture of Moshkovitz and Shapira. A key ingredient in our proof is the construction of a normalized matching flow on the cover graph of the poset $[t]^n$ in which the distribution of weights is close to uniform, a result that may be of independent interest.