On Lih's conjecture concerning Spernerity
type: Item
instance of: scholarly article
title: On Lih's conjecture concerning Spernerity
zbMATH Open document ID: 0916.05068
author: David G. C. Horrocks
published in: European Journal of Combinatorics
publication date: 1999-07-07T00:00:00Z
review text: \textit{Ko-Wei Lih} proved in 1980 [J. Comb. Theory, Ser. A 29, 182-185 (1980; Zbl 0446.05002)] that if \(Y\) is a fixed \(k\)-element subset of \([n]\) and \(P\) denotes the poset that consists of all subsets of \([n]\) having non-empty intersection with \(Y\) then \(P\) satisfies the Sperner property. Lih observed that \(P\) can be considered as a filter in the Boolean algebra \({\mathcal B}_n\) generated by \(k\) one-element sets. From this result he conjectured that if the poset \(P\) is a filter in \({\mathcal B}_n\) generated by a non-empty collection of \(t\)-element subsets, then \(P\) is also Sperner. The conjecture was proved to be false for \(t\geq 4\). This paper proves it for the case \(t=2\).
Mathematics Subject Classification ID: 05D05
zbMATH DE Number: 1275379
zbMATH Keywords: Sperner theorem
zbMATH Keywords: Lih's theorem
zbMATH Keywords: poset filters
zbMATH Keywords: intersection
zbMATH Keywords: Sperner property
Wikidata QID: Q123152673
reviewed by: Péter L. Erdős
MaRDI profile type: Publication
OpenAlex ID: W1969178215
DOI: 10.1006/EUJC.1998.0261
description: scientific article; zbMATH DE number 1275379