  We prove three theorems giving extremal bounds on the incidence structures
determined by subsets of the points and blocks of a balanced incomplete block
design (BIBD). These results generalize and strengthen known bounds on the
number of incidences between points and $m$-flats in affine geometries over
finite fields. First, we show an upper bound on the number of incidences
between sufficiently large subsets of the points and blocks of a BIBD. Second,
we show that a sufficiently large subset of the points of a BIBD determines
many t-rich blocks. Third, we show that a sufficiently large subset of the
blocks of a BIBD determines many t-rich points. These last two results are new
even in the special case of incidences between points and $m$-flats in an
affine geometry over a finite field.
  As a corollary we obtain a tight bound on the number of t-rich points
determined by a set of points in a plane over a finite field, and use it to
sharpen a result of Iosevich, Rudnev, and Zhai on the number of triangles with
distinct areas determined by a set of points in a plane over a finite field.
