$R(5,5)\le 46$

TL;DR

This paper establishes an upper bound of 46 for the Ramsey number R(5,5) using a combination of linear programming and extensive computer case checking, confirming the bound through independent implementations.

Contribution

It provides a new upper bound for R(5,5) and demonstrates a computational approach with independent verification for mathematical proof.

Findings

R(5,5) ≤ 46 established

Computational methods verified independently

Combination of linear programming and case analysis used

Abstract

We prove that the Ramsey number $R(5,5)$ is less than or equal to~$46$. The proof uses a combination of linear programming and checking a large number of cases by computer. All of the computations were independently implemented by both authors, with consistent results.