There is No McLaughlin Geometry

TL;DR

This paper proves that a specific partial geometry with parameters (4,27,2), linked to the McLaughlin graph, does not exist, using symmetry and distributed computing techniques.

Contribution

It resolves a 40-year open problem by demonstrating the non-existence of a particular partial geometry with given parameters.

Findings

No partial geometry with parameters (4,27,2) exists.

A pseudogeometric strongly regular graph can achieve Krein bound equality without being a partial geometry.

The proof employs symmetry and high-performance distributed computing.

Abstract

We determine that there is no partial geometry ${\cal G}$ with parameters $(s,t,\alpha)=(4,27,2)$. The existence of such a geometry has been a challenging open problem of interest to researchers for almost 40 years. The particular interest in ${\cal G}$ is due to the fact that it would have the exceptional McLaughlin graph as its point graph. Our proof makes extensive use of symmetry and high-performance distributed computing, and details of our techniques and checks are provided. One outcome of our work is to show that a pseudogeometric strongly regular graph achieving equality in the Krein bound need not be the point graph of any partial geometry.