Random and deterministic versions of extremal poset problems

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2016-01-01
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Hogenson, Kirsten
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Ryan R. Martin
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Mathematics
Abstract

Let P(n) denote the set of all subsets of {1,...,n} and let P(n,p) be the set obtained from P(n) by selecting elements independently at random with probability p. The Boolean lattice is a partially ordered set, or poset, consisting of the elements of P(n), partially ordered by set inclusion. A basic question in extremal poset theory asks the following: Given a poset P, how big is the largest family of sets in the Boolean lattice which does not contain the structure P as a subposet? The following random analogue of this question is also of interest: Given a poset P, how big is the largest family of sets in P(n,p) which does not contain the structure P as a subposet? In this thesis, we present new proofs for a collection of deterministic extremal subposet problems. We also discuss a new technique called the Hypergraph Container Method in depth and use it to prove a random version of De Bonis and Katona's (r+1)-fork-free theorem.

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Fri Jan 01 00:00:00 UTC 2016