New Upper and Lower Bounds on the Rado Numbers

TL;DR

This paper improves bounds on Rado numbers for linear equations under multiple colorings and uses computational methods to find new Rado numbers, advancing understanding of these combinatorial constants.

Contribution

It provides improved upper and lower bounds for Rado numbers and introduces computational techniques to discover new Rado numbers.

Findings

Established better upper bounds for Rado numbers

Derived new lower bounds for specific equations and colorings

Discovered many new Rado numbers using computational methods

Abstract

If E is a linear homogenous equation and c a natural then the Rado number $R_c(E)$ is the least N so that any c-coloring of the positive integers from 1 to N contains a monochromatic solution. Rado characterized for which E R_c(E) always exists. The original proof of Rado's theorem gave enormous bounds on R_c(E) (when it existed). In this paper we establish better upper bounds, and some lower bounds, for R_c(E) for some c and E. In the appendix we use some of our theorems, and ideas from a probabilistic SAT solver, to find many new Rado Numbers.