Combinatorics of finite sets

Let $\lbrack n\rbrack = \{1,2,\..., n\},A$ and let $2\sp{\lbrack n\rbrack}$ represent the subset lattice of (n) with sets ordered by inclusion. A collection I of subsets of (n) is called an ideal if every subset of a member of I is also in I. An intersecting family S in 2$\sp{\lbrack n\rbrack }$ is called a star if there exists an element of (n) belonging to every member of S, and it is a 1-star if the intersection of every two members of I is exactly that element. Chvatal conjectured that if I is any ideal, then among the intersecting subfamilies of I of maximum cardinality there is a star. In Chapter 1, we prove Chvatal's conjecture for several special cases. Let I be an ideal in 2$\sp{\lbrack n\rbrack }$ that is compressed with respect to a given element. We prove that among the largest intersecting families of I there is a star. We also prove that if the maximal elements $B\sb1,\...,B\sb{q}$ of an ideal I can be partitioned into two 1-stars, then I satisfies Chvatal's conjecture.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Snevily, Hunter Saint Clair

Contributors dc:contributor

• Reznick, Bruce

• Mathematics

Statement dc:rights

• Copyright 1991 Snevily, Hunter Saint Clair

Language dc:language

eng

Identifier

AAI9210996
(UMI)AAI9210996

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/21100