Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as H|S = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VC-dimension) of H is the size of the largest subset S for which H|S has 2 |S| edges. Hypergraphs of small VC-dimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.

Let X be a finite point set in the plane. We consider the set system on X whose sets are all intersections of X with a halfplane. Similarly one can investigate set systems defined on point sets in higher-dimensional spaces by other classes of simple geometric figures (simplices, balls, ellipsoids, etc.). It turns out that simple combinatorial properties of such set systems (most notably the Vapnik-Chervonenkis dimension and related concepts of shatter functions) play an important role in several areas of mathematics and theoretical computer science. Here we concentrate on applications in discrepancy theory, in combinatorial geometry, in derandomization of geometric algorithms, and in geometric range searching. We believe that the described tools might be useful in other areas of mathematics too. 1 Introduction For a set system S ` 2 X on an arbitrary ground set X and for A ` X, we write Sj A = fS " A; S 2 Sg for the set system induced by S on A (or the trace of S on A). Let H den...

We consider asymptotic bounds for the discrepancy of point sets on a class of fractal sets. By a method of R. Alexander, we prove that for a wide class of fractals, the L 2 -discrepancy (and consequently also the worst-case discrepancy) of an N-point set with respect to halfspaces is at least of the order N^(-1/2-1/(2s)) , where s is the Hausdorff dimension of the fractal. We also show that for many fractals, this bound is tight for the L 2 -discrepancy. Determining the correct order of magnitude of the worst-case discrepancy remains a challenging open problem. Keywords: discrepancy, fractals, halfspaces