#### Date of Award

2019

#### Degree Name

Mathematics

#### College

College of Science

#### Type of Degree

M.A.

#### Document Type

Thesis

#### First Advisor

Dr. Michael Schroeder, Committee Chairperson

#### Second Advisor

Dr. JiYoon Jung

#### Third Advisor

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 B_{1}, B_{2}, . . . , B_{t} such that (a) |B_{1}| = |B_{2}| = · · · = |B_{t} | = m, (b) the largest element in Bi is less than the smallest element in B_{i}+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.

#### Recommended Citation

O’Neal, Adam, "On monochromatic sets with nondecreasing diameter" (2019). *Theses, Dissertations and Capstones*. 1238.

https://mds.marshall.edu/etd/1238