A tight lower bound for the hardness of clutters

07/17/2018

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by   Vahan Mkrtchyan, et al.

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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 s_e of an edge e of a clutter is recognizing for e, if s_e 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 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.