Small tight sets in finite elliptic, parabolic and Hermitian polar spaces

Abstract

Small tight sets have been classified in finite classical polar spaces of symplectic type, hyperbolic type and one of the two Hermitian types [1]. The approach used is not working for the remaining finite classical polar spaces, mainly due to the fact that a tight set for these is not necessarily a minihyper, so the known results on blocking sets can not be applied. This might be the reason that there is no classification for small tight sets in these spaces. This paper provides a classification of tight sets with parameter x in these spaces provided that x is small compared to the order q of the polar space. One of the results is that a tight set of the generalized quadrangle Q(4,q) that is not the union of disjoint lines has at least ($$\sqrt q $$q + 1)(q + 1) points with equality if and only if it consists of the points of an embedded subquadrangle of order $$\sqrt q $$q.