Note on the number of antichains in generalizations of the Boolean lattice

Abstract

We give a short and self-contained argument that shows that, for any positive integers $t$ and $n$ with $t =O\Bigl(\frac{n}{\log n}\Bigr)$, the number $\alpha([t]^n)$ of antichains of the poset $[t]^n$ is at most \[\exp_2\Bigl(1+O\Bigl(\Bigl(\frac{t\log^3 n}{n}\Bigr)^{1/2}\Bigr)\Bigr)N(t,n)\,,\] where $N(t,n)$ is the size of a largest level of $[t]^n$. This, in particular, says that if $t \ll n/\log^3 n$ as $n \rightarrow \infty$, then $\log\alpha([t]^n)=(1+o(1))N(t,n)$, giving a (partially) positive answer to a question of Moshkovitz and Shapira for $t, n$ in this range. Particularly for $t=3$, we prove a better upper bound: \[\log\alpha([3]^n)\le(1+4\log 3/n)N(3,n),\] which is the best known upper bound on the number of antichains of $[3]^n$.