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.

We show for k � 2 that if q � 3 and n � 2k + 1, or q = 2 and n � 2k + 2, then any intersecting family F of k-subspaces of an n-dimensional vector space over GF(q) with ⋂ F ∈F F = 0 has size at most [] n−1 k−1 − qk(k−1) [] n−k−1 k−1 + qk. This bound is sharp as is shown by Hilton-Milner type families. As an application of this result, we determine the chromatic number of the corresponding q-Kneser graphs.