## Electronic Theses and Dissertations

#### Document Type

Thesis - Open Access

2017

#### Degree Name

Master of Science (MS)

#### Department / School

Mathematics and Statistics

Daniel Schaal

Donald Vestal

#### Abstract

For every positive integer $a$, let $n = R_{ZS}(a)$ be the least integer, provided it exists, such that for every coloring $\Delta : \{1, 2, ..., n\} \rightarrow \{0, 1, 2\},$ there exist three integers $x_1, x_2, x_3$ (not necessarily distinct) such that $\Delta(x_1) + \Delta(x_2) + \Delta(x_3) \equiv 0\ (mod\ 3)$ and $ax_1 +x_2 = x_3.$ If such an integer does not exist, then $R_{ZS}(a) = \infty.$ The main results of this paper are $R_{ZS}(2) = 12$ and a lower bound is found for $R_{ZS}(a)$ where $a \geq 2$.

#### Library of Congress Subject Headings

Ramsey theory.
Combinatorial analysis.

#### Description

Includes bibliographical references (page 16)

application/pdf

20

#### Publisher

South Dakota State University

COinS