Bounds on maximal families of sets not containing three sets with \(A\cap B \subset C\), \(A \not\subset B\)
type: Item
instance of: scholarly article
title: Bounds on maximal families of sets not containing three sets with \(A\cap B \subset C\), \(A \not\subset B\)
zbMATH Open document ID: 1171.05051
author: Teena Carroll
author: Gyula O. H. Katona
published in: Order
publication date: 2008-11-17T00:00:00Z
review text: Let \(\text{La}^\sharp(n,V_2)\) denote the size of the largest family \(\mathcal{F}\) of subsets of \(\{1,2,\dots,n\}\) containing no three distinct members \(F, G_1, G_2 \in \mathcal{F}\) such that \(F \subset G_1\), \(F \subset G_2\), \(G_1 \not\subset G_2\). The following inequalities are proved in this paper: \[ {n \choose {\lfloor \frac{n}{2} \rfloor}} \left(1 + \frac{1}{n} + \Omega \left(\frac{1}{n^2}\right)\right)\leq\text{La}^\sharp(n,V_2)\leq {n \choose {\lfloor \frac{n}{2} \rfloor}}\left(1 + \frac{2}{n} + O\left(\frac{1}{n^2}\right)\right). \]
Mathematics Subject Classification ID: 05D05
zbMATH DE Number: 5366786
zbMATH Keywords: extremal problem for families
zbMATH Keywords: Sperner type theorem
zbMATH Keywords: forbidden subposet
MaRDI profile type: Publication
full work available at URL: s11083-008-9090-8
OpenAlex ID: W2083743655
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DOI: 10.1007/S11083-008-9090-8
description: scientific article; zbMATH DE number 5366786