The author’s of this blog have recently submitted a paper together: “Every flock generalised quadrangle has a hemisystem”. A hemisystem of a generalised quadrangle of order (s,t) is a set H of lines such that each point P lies on (t+1)/2 lines of H; that is, half of the lines incident with P lie within H. In particular, a hemisystem of a generalised quadrangle of order $(q^2,q)$ would yield a partial quadrangle and hence a strongly regular graph. The only known generalised quadrangles of order $(q^2,q)$, q odd, are those arising from the flock/q-clan construction; the flock generalised quadrangles. The points and lines of the 3-dimensional Hermitian variety are the classical flock quadrangles, and Beniamino Segre introduced the notion of hemisystems in his study of regular systems of this GQ. In fact, Segre showed (in 1965) that $H(3,3^2)$ has a hemisystem, that it was unique and that the Gewirtz graph arose as the concurrency graph of this object.
For thirty years, nobody could construct another hemisystem of $H(3,q^2)$, and Jef Thas conjectured in the Handbook of Incidence Geometry (1995) that no hemisystems of $H(3,q^2)$ exist for $q>3$. Ten years later, Tim Penttila and Antonio Cossidente (2005) construct for each odd q a hemisystem of $H(3,q^2)$, each arising from an embedded elliptic quadric $Q^-(3,q)$. We show in our paper, that in fact, every flock generalised quadrangle (with an even number of lines) contains a hemisystem. We give a construction which is analogous to the Penttila-Cossidente construction. That is, we perturb their example in a tricky way…