Symbolic Sets for Proving Bounds on Rado Numbers

2025-05-17

AI Summary

This paper determines exact values and asymptotic bounds for the 3-color Rado numbers $ R_3(ax + by = bz) $ and $ R_3(ax + ay = bz) $. Addressing two previously unresolved classes of linear Diophantine equations, we propose a novel methodology integrating SAT solving with symbolic set reasoning: we design a custom verification framework supporting parametric enumeration of set intersections and unions, and automate case analysis via integration of SymPy and Z3. Furthermore, leveraging structural patterns in SAT solutions, we derive a generalizable constructive paradigm for upper bounds, overcoming limitations of manual derivation. Our approach yields the first exact computations of multiple 3-color Rado numbers and establishes tight $ Theta(b) $ asymptotic upper bounds for both equation families. All results are fully machine-verified.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Computing unknown Rado numbers for specific linear equations
• Proving general bounds on Rado numbers using SAT solvers
• Developing a tool for symbolic set operations in proofs

Innovation

• Using SAT solvers for Rado numbers computation
• Automated proof checking via symbolic sets
• Developing a tool with SymPy and Z3