Elsevier

Discrete Mathematics

Volumes 208–209, 28 October 1999, Pages 589-605
Discrete Mathematics

Finite generalized quadrangles as the union of few large subquadrangles

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Abstract

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.

Keywords

Finite generalized quadrangles
Subquadrangles
Polar spaces
Polar subspaces

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1

The author is a Research Director of the Fund for Scientific Research, Flanders, Belgium.