Ryan Wood

Year of Award

2024

Document Type

Dissertation

Degree Type

Doctor of Philosophy (PhD)

Degree Name

Mathematics

Department or School/College

Department of Mathematical Sciences

Committee Chair

P. Mark Kayll

Committee Co-chair

Cory Palmer

Commitee Members

Kelly McKinnie, Fred Peck, Armond Duwell

Keywords

Extremal Combinatorics, Extremal Set Theory

Abstract

In this work, we present results employing the flower base method. We work in the field of extremal set theory, focusing our attention on r-uniform intersecting families. The classic Erd˝os-Ko-Rado Theorem states that, for n large enough, the largest runiform intersecting family, a so-called maximum star, consists of taking all r-sets containing a fixed element. Lemons and Palmer [31] investigated the diversity function, showing that the largest intersecting family that is the most “un-star-like” consists of taking all r-sets containing at least two members of a set of size three. Our main problem, that of C-weighted diversity, unifies these two results as solutions for different ranges of a weight C. In fact, we characterize the extremal families for all C < 7/3. We also use flower bases to prove a stability result about diversity and reprove another stability result due to Kupavskii and Zakharov [30]. Further, we present a survey of base-type methods culminating in a detailed discussion about the flower base.

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