# Hemisystems of flock generalised quadrangles

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 would yield a partial quadrangle and hence a strongly regular graph. The only known generalised quadrangles of order , *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 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 , and Jef Thas conjectured in the Handbook of Incidence Geometry (1995) that no hemisystems of exist for . Ten years later, Tim Penttila and Antonio Cossidente (2005) construct for each odd *q* a hemisystem of , each arising from an embedded elliptic quadric . 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…