## Mathematics, Department of

#### First Advisor

Jamie Radcliffe

#### Date of this Version

Summer 8-2017

#### Abstract

In this thesis, we present a number of results, mostly concerning set systems that are antichains and/or have bounded diameter. Chapter 1 gives a more detailed outline of the thesis. In Chapter 2, we give a new short proof of Kleitman's theorem concerning the maximal size of a set system with bounded diameter. In Chapter 3, we turn our attention to antichains with bounded diameter. Šileikis conjectured that an antichain of diameter *D* has size at most (*n/D*/2). We present several partial results towards the conjecture.

In 2014, Leader and Long gave asymptotic bounds on the size of a set system where |*A\B*| ≠ 1 and more generally, when |*A\B*| ≠ *k*. In Chapter 4, we present streamlined versions of their proofs, with slightly better bounds.

The final chapter presents a proof for the following poset analog of an elementary graph theory problem: every poset with |*R*| relations contains a height two subposet with at least |*R*|/2 relations.

## Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Jamie Radcliffe. Lincoln, Nebraska: August, 2017

Copyright (c) 2017 Brent McKain