Optimal Embeddings of Posets in Hypercubes

TL;DR

The paper addresses the problem of embedding finite posets into hypercubes, focusing on the hypercube-width $w^*(\mathcal{P})$. It proves the conjecture $w^*(\mathcal{P})\le |\mathcal{P}|$ for all finite posets by first treating two-layered posets and then extending to general posets using Hall's theorem to refine an induced copy into a controlled, low-height embedding. A key technique is constructing matchings in bipartite graphs to replace certain minimal elements by singletons while preserving the poset order, ensuring the embedding remains within height $h^*(\mathcal{P})$. As a consequence, the work improves saturation-number bounds to $\text{sat}^*(n,\mathcal{P})\le 2n^{|\mathcal{P}|-1}$ for large $n$, and it discusses equality cases $w^*(\mathcal{P})=|\mathcal{P}|$ along with open structural questions linking hypercube-height and hypercube-width.

Abstract

Given a finite poset $\mathcal P$, the hypercube-height, denoted by $h^*(\mathcal P)$, is defined to be the largest $h$ such that, for any natural number $n$, the subsets of $[n]$ of size less than $h$ do not contain an induced copy of $\mathcal P$. The hypercube-width, denoted by $w^*(\mathcal P)$, is the smallest $w$ such that the subsets of $[w]$ of size at most $h^*(\mathcal P)$ contain an induced copy of $\mathcal P$. In other words, $h^*(\mathcal P)$ asks how `low' can a poset be embedded, and $w^*(\mathcal P)$ asks for the first hypercube in which such an `optimal' embedding occurs. These notions were introduced by Bastide, Groenland, Ivan and Johnston in connection to upper bounds for the poset saturation numbers. While it is not hard to see that $h^*(\mathcal P)\leq |\mathcal P|-1$ (and this bound can be tight), the hypercube-width has proved to be much more elusive. It was shown by the authors mentioned above that $w^*(\mathcal P)\leq|\mathcal P|^2/4$, but they conjectured that in fact $w^*(\mathcal P)\leq |\mathcal P|$ for any finite poset $\mathcal P$. In this paper we prove this conjecture. The proof uses Hall's theorem for bipartite graphs as a precision tool for modifing an existing copy of our poset.

Theorems & Definitions (13)

• Conjecture 1: Conjecture 9 in polynomial
• Theorem 2: Hall
• Theorem 3
• proof
• Claim A
• proof
• Claim B
• proof
• Theorem 4
• proof