Journal of Statistical Planning Intersection-sets in Pg(n,2)i

We want to examine line-free subsets of projective spaces over GF(2) especially from an intersection-set point of view. In this paper, we describe incidence equations for such sets. For every n~>3, we construct a line-free subset (or cap), X,, of PG(n,2), which meets every hyperplane in at least 2" 3 points. We make use of the incidence equations and other methods to show that, remarkably, up to isomorphism, the set Xn is the unique set having these two properties for n = 3, 4, 5. The cases n = 3,4 are small enough to handle directly. The result for n -5 was in fact established in Fugrre et al. (J. Combin. Designs 2 (5) (1994) 287-299) using a labourious, detailed case by case analysis. One of the motivations for that result was to decide the chromatic number of PG(5,2). We also establish various sporadic results on r-intersection-sets in PG(n,2) when n = 5,6. A M S classification: primary 51 E 15; secondary 05 B 10