Extremal hypergraph problems and convex hulls
type: Item
instance of: scholarly article
title: Extremal hypergraph problems and convex hulls
zbMATH Open document ID: 0579.05033
DOI: 10.1007/BF02579438
author: Peter Frankl
author: Péter L. Erdős
author: Gyula O. H. Katona
published in: Combinatorica
publication date: 1985-01-01T00:00:00Z
review text: Let X be a finite set of n elements and F be a family of its subsets. Then \(F_ k\) denotes the subfamily of the k-element subsets in F and \(| F_ k| =f_ k\). The vector \((f_ 0,f_ 1,...,f_ n)\) in the \((n+1)\)-dimensional Euclidean space \(R^{n+1}\) is called the profile of F. If c is a finite set in \(R^{n+1}\), the convex hull of c is the set of all convex linear combinations of the elements of c. \(p\in c\) is an extreme point of c iff p is not a convex linear combination of elements of c different from p. The determination of the convex hull of a set is equivalent to finding its extreme points. k-Sperner-family has no \(k+1\) different members satisfying \(A_ 1\subset...\subset A_{k+1}\). A family F is intersecting if A,B\(\in F\) implies \(A\cap B\neq \emptyset.\) The present paper tries to determine the extreme points of the set of profiles of the simplest known classes of families. The effort is successful for 3 classes: (1) intersecting families, (2) k-Sperner- families, (3) \(G_ 1,...,G_ t\) are not necessarily disjoint families, where \(A\in G_ i\), \(B\in G_ j\), \(i\neq j\), \(A\neq B\) imply \(A\not\subset B\). The results contain many old theorems of extremal set theory as particular cases (Sperner, Erdős-Ko-Rado, Daykin-Frankl- Green-Hilton).
Mathematics Subject Classification ID: 05C35
Mathematics Subject Classification ID: 05C65
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 3926951
zbMATH Keywords: family of subsets
zbMATH Keywords: profile
zbMATH Keywords: convex hull
zbMATH Keywords: extreme points
reviewed by: B. Andrasfai
MaRDI profile type: Publication
cites work: Q3942966
cites work: On a lemma of Littlewood and Offord
cites work: INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS
cites work: Intersecting Sperner families and their convex hulls
cites work: Intersection theorems for systems of finite sets
cites work: A simple proof of the Erdős-Chao Ko-Rado theorem
full work available at URL: https://doi.org/10.1007/bf02579438
OpenAlex ID: W2028845083
description: scientific article; zbMATH DE number 3926951