Induced poset saturation in the hypergrid

Abstract

Set $[n]=\{1, 2, \ldots , n\}$. The hypergrid $[t]^n$ is the collection of functions $f: \ [n]\rightarrow [t]$. We equip it with the natural partial order by letting $f\leq g$ whenever $f(x)\leq g(x)$ holds for all $x\in [n]$. Given a poset $P$ which can be embedded as an induced subposet of $[t]^n$, the induced poset saturation function $\mathrm{sat}^{\star}([t]^n, P)$ denotes the minimum size of a subset of $[t]^n$ that is both induced $P$-free and induced $P$-saturated. We show that for all $t\geq 2$, $\mathrm{sat}^{\star}([t]^n, P)$ satisfies a dichotomy: for every poset $P$, either there exists a constant $C_P$ such that $\mathrm{sat}^{\star}([t]^n, P)=C_P$ for all $n$ sufficiently large, or $\mathrm{sat}^{\star}([t]^n, P)=\Omega(\sqrt{n})$. We also show chains fall in the former part of the dichotomy, while posets with the unique twin cover property fall in the latter part. These contributions generalize a number of results obtained by various authors in the hypercube ($t=2$) setting; the transition to the hypergrid setting provides novel challenges, however, and requires some new ideas.