We show that small blocking sets in PG(n, q) with respect to hyperplanes intersect every hyperplane in 1 modulo p points, where q= p h . The result is then extended to blocking sets with respect to k-dimensional subspaces and, at least when p>2, to intersections with arbitrary subspaces not just hyperplanes. This can also be used to characterize certain non-degenerate blocking sets in higher dimensions. Furthermore we determine the possible sizes of small minimal blocking sets with respect to k-dimensional subspaces.

The spectrum problem for minimal blocking sets means that we wish to determine the possible cardinalities of minimal blocking sets. Besides surveying the results on this problem some new results (or new proofs) are given.

The size of large minimal blocking sets is bounded by the Bruen-Thas upper bound. The bound is sharp when q is a square. Here the bound is improved if q is a non-square. On the other hand, we present some constructions of reasonably large minimal blocking sets in planes of non-prime order. The construction can be regarded as a generalization of Buekenhout's construction of unitals. For example, if q is a cube, then our construction gives minimal blocking sets of size q 4=3 þ 1 or q 4=3 þ 2. Density results for the spectrum of minimal blocking sets in Galois planes of non-prime order is also presented. The most attractive case is when q is a square, where we show that there is a minimal blocking set for any size from the interval ½4q log q; q ffiffi ffi q p À q þ 2 ffiffi ffi q p . #

Abstract. In this survey recent results about q-analogues of some classical theorems in extremal set theory are collected. They are related to determining the chromatic number of the q-analogues of Kneser graphs. For the proof one needs results on the number of 0-secant subspaces of point sets, so in the second part of the paper recent results on the structure of point sets having few 0-secant subspaces are discussed. Our attention is focussed on the planar case, where various stability results are given.Mathematics Subject Classification (2010). 05D05, 05B25.

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