Small maximal partial ovoids in generalized quadrangles

Abstract

A {\em maximal partial ovoid} of a generalized quadrangle is a maximal set of points no two of which are collinear. The problem of determining the smallest size of a maximal partial ovoid in quadrangles has been extensively studied in the literature. In general, theoretical lower bounds on the size of a maximal partial ovoid in a quadrangle of order $(s,t)$ are linear in $s$. In this paper, in a wide class of quadrangles of order $(s,t)$ we give a construction of a maximal partial ovoid of size at most $s \cdot \mbox{polylog}(s)$, which is within a polylogarithmic factor of theoretical lower bounds. The construction substantially improves previous quadratic upper bounds in quadrangles of order $(s,s^2)$, in particular in the well-studied case of the elliptic quadrics $Q^-(5,s)$.