A dominating set in a graph is a set of vertices such that each vertex not in the set has a neighbor in the set. The domination number is the smallest size of a dominating set. We consider this problem in the incidence graph of a generalized quadrangle. We show that the domination number of a generalized quadrangle with parameters s and t is at most 2st + 1, and we prove that this bound is sharp if s = t or if s = q − 1 and t = q + 1. Moreover, we give a complete classification of smallest dominating sets in generalized quadrangles where s = t, and give some general results for small dominating sets in the general case.

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A (k, k − t)-SCID (set of Subspaces with Constant Intersection Dimension) is a set of k-dimensional vector spaces that have pairwise intersections of dimension k − t. Let C = {π 1 , . . . , πn} be a (k, k − t)-SCID. Define S := π 1 , . . . , πn and I := π i ∩ π j | 1 ≤ i < j ≤ n . We establish several upper bounds for dim S + dim I in different situations. We give a spectrum result under certain conditions for n, giving examples of (k, k − t)-SCIDs reaching a large interval of values for dim S + dim I.

A (k, k − t)-SCID (set of Subspaces with Constant Intersection Dimension) is a set of k-dimensional vector spaces that have pairwise intersections of dimension k − t. Let C = {π 1 , . . . , π n } be a (k, k − t)-SCID. Define S := π 1 , . . . , π n and I := π i ∩ π j | 1 ≤ i < j ≤ n . We establish several upper bounds for dim S +dim I in different situations. We give a spectrum result for the case (n − 1)(k − t) ≤ k and for the case n ≤ q t(n−η) −1 q t −1 , giving examples of (k, k −t)-SCIDs reaching a large interval of values for dim S + dim I.