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arXiv · 2605.00622

Sizes of witnesses in Covtree

Jette Gutzeit·Kimia Shaban·Karen Yeats·Stav Zalel
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Abstract

Given a set $\Gamma$ of $k$ unlabelled posets, each of size $n$, we say that a poset $Q$ is a \emph{witness} to $\Gamma$ if $\Gamma$ is the set of downsets of size $n$ of $Q$. We say that $Q$ is a \emph{minimal witness} if it does not contain a proper downset that is itself a witness to $\Gamma$. Motivated by the causal set approach to quantum gravity, we study the upper bound on the size of minimal witnesses as a function of $n$ and $k$. We show that there is no linear upper bound of the form $n+k+c$ for any constant $c$. We introduce the \emph{exchange graph of downsets} as a new tool to study this scenario, and use it to show that all minimal witnesses $Q$ satisfy the bound $|Q|\leq nk-n$, and that when $k=3$ there is at least one minimal witness $Q$ that satisfies the bound $|Q|\leq \frac{3}{2}(n+1)$.

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BibTeXRIS

Jette Gutzeit, Kimia Shaban, Karen Yeats, Stav Zalel. 2026-05-01. Sizes of witnesses in Covtree. https://arxiv.org/abs/2605.00622

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