On Zero-Sum Rado Numbers for the Equation ax_1 + x_2 = x_3

Author

Nicholas Brown, South Dakota State UniversityFollow

Document Type

Thesis - Open Access

Award Date

2017

Degree Name

Master of Science (MS)

Department / School

Mathematics and Statistics

First Advisor

Daniel Schaal

Second Advisor

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)

Format

application/pdf

Number of Pages

20

Publisher

South Dakota State University

Recommended Citation

Brown, Nicholas, "On Zero-Sum Rado Numbers for the Equation ax_1 + x_2 = x_3" (2017). Electronic Theses and Dissertations. 1719.
https://openprairie.sdstate.edu/etd/1719