A tight lower bound for the hardness of clutters

TL;DR

This paper establishes a tight lower bound on the hardness of clutters, a combinatorial structure, demonstrating that the bound is asymptotically optimal through an infinite sequence of clutters.

Contribution

It provides the first asymptotically optimal lower bound on the hardness of any clutter, advancing understanding of their combinatorial complexity.

Findings

Established a tight lower bound for clutter hardness

Proved the bound is asymptotically best possible

Constructed an infinite sequence of clutters attaining the bound

Abstract

A {\it clutter} (or {\it antichain} or {\it Sperner family}) $L$ is a pair $(V,E)$, where $V$ is a finite set and $E$ is a family of subsets of $V$ none of which is a subset of another. Normally, the elements of $V$ are called {\it vertices} of $L$, and the elements of $E$ are called {\it edges} of $L$. A subset $s_e$ of an edge $e$ of a clutter is {\it recognizing} for $e$, if $s_e$ is not a subset of another edge. The {\it hardness} of an edge $e$ of a clutter is the ratio of the size of $e\textrm{'s}$ smallest recognizing subset to the size of $e$. The hardness of a clutter is the maximum hardness of its edges. In this short note we prove a lower bound for the hardness of an arbitrary clutter. Our bound is asymptotically best-possible in a sense that there is an infinite sequence of clutters attaining our bound.