## Theses, Dissertations and Capstones

2019

Mathematics

#### College

College of Science

M.A.

#### Document Type

Thesis

Dr. Michael Schroeder, Committee Chairperson

Dr. JiYoon Jung

Dr. Carl Mummert

#### Abstract

Our problem comes from the field of combinatorics known as Ramsey theory. Ramsey theory, in a general sense, is about identifying the threshold for which a family of objects, associated with a particular parameter, goes from never or sometimes satisfying a certain property to always satisfying that property. Research in Ramsey theory has applications in design theory and coding theory. For integers m, r, and t, we say that a set of n integers colored with r colors is (m, r, t)-permissible if there exist t monochromatic subsets B1, B2, . . . , Bt such that (a) |B1| = |B2| = · · · = |Bt | = m, (b) the largest element in Bi is less than the smallest element in Bi+1 for 1 ≤ i ≤ t − 1, and (c) the diameters of the subsets are nondecreasing. We define f(m, r, t) to be the smallest integer n such that every string of length n is (m, r, t)-permissible. In this thesis, we first look at some preliminary results for values of f(m, r, t), specifically when each individual parameter is 1 as the others vary. We then show that f(m, r, t) exists for all possible positive parameters. We proceed by determining f(2, 2, t) for all positive integers t. We conclude by considering colorings with more than two colors and monochromatic sets that have more than 2 elements, as well as investigating an enumeration of the number of ways a string could be realized as (m, r, t)-permissible.

#### Subject(s)

Ramsey theory.

Combinatorial analysis.

COinS