## Document Type

Thesis - University Access Only

## Award Date

2000

## Degree Name

Master of Science (MS)

## Department / School

Mathematics and Statistics

## Abstract

An equation L is said to have a finite 4-color Rado number if there exists a least integer n, such that every 4-coloring of the integers {J I 1 ≤ j ≤ n} has an ordered triple (x_{1 },x_{2} ,x_{3 }) such that (x_{1 },x_{2} ,x_{3 }) is a monochromatic solution to L. A particular pattern, the stubborn pattern, is used to provide some framework in the discussion. This pattern is used to prove one of the two general theorems creating upper and lower bounds on the Rado number for the equation x_{1} + x_{2} + c = x_{3} where c is any nonnegative integer. We represent the Rado number for the equation x_{1} + x_{2} + c = x_{3} by R_{t} (c) where tis the number of colors used and c is the constant c in the equation. An exhaustive computer search is performed to obtain exact 4-color Rado numbers for 3 integers, specifically, R_{4} (1) = 83, R_{4} (2) = 121, and R_{4} (3) = 161. This implies exact values for half of the positive integers, specifically, for all c = 2 (mod3) and c = 3 (mod4). Upper and lower bounds are presented and a conjecture is made regarding the remaining integers.

## Library of Congress Subject Headings

Combinatorial analysis Rado numbers

## Format

application/pdf

## Number of Pages

79

## Publisher

South Dakota State University

## Recommended Citation

Malo, Robert, "Four Color Rado Numbers for the Equation x₁+x₂+c=x₃" (2000). *Electronic Theses and Dissertations*. 760.

https://openprairie.sdstate.edu/etd2/760