An inequality for generalized quadrangles

Let S be a generalized quadrangle of order (s, t). Let X and Y be disjoint sets of pairwise noncollinear points of S such that each point of X is coUinear with each point of Y. If m = \X\ and n = | Y\, then (m — l)(n - 1) 1, t > 1, is a point-line incidence geometry S = (??, £, 7) with point set 1, t > 1. Let X = (xx, . . . , xm) and Y = (y\> ■ • • >yn) De disjoint sets of pairwise noncollinear points of §, m > 2 and n > 2. Let ki be the number of x/s with which v, is collinear, 1 < i < n, 0 < k, < m. Our main results consist of the following two theorems.