Intriguing sets of partial quadrangles

TL;DR

This paper extends the theory of intriguing sets from generalized quadrangles to partial quadrangles, revealing new structural insights into hemisystems and related intriguing sets.

Contribution

It generalizes the concept of intriguing sets to partial quadrangles, providing new understanding of their structure and connections to generalized quadrangles.

Findings

Extended intriguing set theory to partial quadrangles.

Gained insights into hemisystems and intriguing sets.

Linked partial quadrangles with generalized quadrangles structures.

Abstract

The point-line geometry known as a \textit{partial quadrangle} (introduced by Cameron in 1975) has the property that for every point/line non-incident pair $(P,\ell)$, there is at most one line through $P$ concurrent with $\ell$. So in particular, the well-studied objects known as \textit{generalised quadrangles} are each partial quadrangles. An \textit{intriguing set} of a generalised quadrangle is a set of points which induces an equitable partition of size two of the underlying strongly regular graph. We extend the theory of intriguing sets of generalised quadrangles by Bamberg, Law and Penttila to partial quadrangles, which surprisingly gives insight into the structure of hemisystems and other intriguing sets of generalised quadrangles.