Finite generalized quadrangles as the union of few large subquadrangles Original Research Article

We study the question: what is the smallest number n of subquadrangles of order (s,t′) of a finite generalized quadrangle Γ of order (s,t) such that the union of the point sets of all these subquadrangles is equal to the point set of Γ? It turns out that n⩾s+1 and if n=s+1, then except for a finite list of small examples, either all the subquadrangles are disjoint, or t=s=t′ and all the subquadrangles meet pairwise in a common subquadrangle of order (s,1). Examples exist in both cases and they show that a further classification is out of reach. A similar result holds for finite polar spaces.