Intriguing sets of partial quadrangles

The point-line geometry known as a $\mathit{\text{partial}}\mathit{\text{quadrangle}}$ (introduced by Cameron in 1975) has the property that for every point/line non-incident pair $(P,\mathrm{\ell })$, there is at most one line through $P$ concurrent with $\mathrm{\ell }$. So in particular, the well-studied objects known as $\mathit{\text{generalised}}\mathit{\text{quadrangles}}$ are each partial quadrangles. An $\mathit{\text{intriguing}}\mathit{\text{set}}$ 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.