A tight lower bound for the hardness of clutters

A clutter (or antichain or 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 vertices of L, and the elements of E are called edges of L. A subset se\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_e$$\end{document} of an edge e of a clutter is recognizing for e, if se\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_e$$\end{document} is not a subset of another edge. The hardness of an edge e of a clutter is the ratio of the size of e's\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e\text {'s}$$\end{document} 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.