# Positive Semidefiniteness of Matrices arising from Ramsey Theory

## TL;DR

This paper proves a conjecture that certain matrices from Ramsey theory are positive semidefinite, enabling more efficient algorithms for bounding small product-Ramsey numbers by explicitly enumerating their eigenvalues and eigenspaces.

## Contribution

It confirms the positive semidefiniteness of matrices related to Ramsey theory and provides explicit eigenstructure analysis using hypergeometric identities.

## Key findings

- Matrices are positive semidefinite, confirming a key conjecture.
- Explicit eigenvalues and eigenspaces are derived.
- Implication for more efficient algorithms in Ramsey number bounds.

## Abstract

We resolve a conjecture of Cooper-Fenner-Purewal that a certain sequence of combinatorial matrices which can be used to bound small product-Ramsey numbers is positive semidefinite. Because the connection to Ramsey Theory involves solving quadratic integer programs associated to these matrices, this implies that there are relatively efficient algorithms for bounding said numbers. The proof is direct, and yields important structural information: we enumerate the eigenvalues and eigenspaces explicitly by employing hypergeometric identities.