Satisfiability and computing van der Waerden numbers

TL;DR

This paper integrates combinatorics and propositional satisfiability to compute van der Waerden numbers, demonstrating that SAT solvers can effectively determine these combinatorial functions and produce new results.

Contribution

It introduces a novel SAT-based approach to compute van der Waerden numbers, bridging combinatorics with satisfiability testing and enabling new computational results.

Findings

SAT techniques are effective for computing van der Waerden numbers

New bounds and values for van der Waerden numbers were obtained

Propositional theories in this context are useful for SAT solver benchmarking

Abstract

In this paper we bring together the areas of combinatorics and propositional satisfiability. Many combinatorial theorems establish, often constructively, the existence of positive integer functions, without actually providing their closed algebraic form or tight lower and upper bounds. The area of Ramsey theory is especially rich in such results. Using the problem of computing van der Waerden numbers as an example, we show that these problems can be represented by parameterized propositional theories in such a way that decisions concerning their satisfiability determine the numbers (function) in question. We show that by using general-purpose complete and local-search techniques for testing propositional satisfiability, this approach becomes effective -- competitive with specialized approaches. By following it, we were able to obtain several new results pertaining to the problem of computing van der Waerden numbers. We also note that due to their properties, especially their structural simplicity and computational hardness, propositional theories that arise in this research can be of use in development, testing and benchmarking of SAT solvers.