Symbolic Sets for Proving Bounds on Rado Numbers

TL;DR

This paper uses SAT solvers and a new symbolic set tool to compute and verify bounds on 3-colour Rado numbers for specific linear equations, advancing understanding in combinatorial number theory.

Contribution

It introduces a novel symbolic set tool and applies SAT solving to determine and verify new bounds on Rado numbers for certain equations.

Findings

Computed new Rado number bounds for specific equations

Developed a symbolic set tool using SymPy and Z3

Automated proof verification for complex case analyses

Abstract

Given a linear equation $\cal E$ of the form $ax + by = cz$ where $a$, $b$, $c$ are positive integers, the $k$-colour Rado number $R_k({\cal E})$ is the smallest positive integer $n$, if it exists, such that every $k$-colouring of the positive integers $\{1, 2, \dotsc, n\}$ contains a monochromatic solution to $\cal E$. In this paper, we consider $k = 3$ and the linear equations $ax + by = bz$ and $ax + ay = bz$. Using SAT solvers, we compute a number of previously unknown Rado numbers corresponding to these equations. We prove new general bounds on Rado numbers inspired by the satisfying assignments discovered by the SAT solver. Our proofs require extensive case-based analyses that are difficult to check for correctness by hand, so we automate checking the correctness of our proofs via an approach which makes use of a new tool we developed with support for operations on symbolically-defined sets -- e.g., unions or intersections of sets of the form $\{f(1), f(2), \dotsc, f(a)\}$ where $a$ is a symbolic variable and $f$ is a function possibly dependent on $a$. No computer algebra system that we are aware of currently has sufficiently capable support for symbolic sets, leading us to develop a tool supporting symbolic sets using the Python symbolic computation library SymPy coupled with the Satisfiability Modulo Theories solver Z3.