On the Smallest Minimal Blocking Sets in Projective Space Generating the Whole Space

It was conjectured that the smallest minimal point sets of PG(2s, q), q a square, that meet every s-subspace and that generate the whole space are Baer subgeometries PG(2s, √ q). This was shown in 1971 by Bruen for s = 1, and by Metsch and Storme [5] for s = 2. Our main interest in this paper is to prepare a possible proof of this conjecture by proving a strong theorem on line-blocking sets in projective spaces (see Theorem 1.1). We apply this theorem to prove the conjecture in the case s = 3. The general case will be handled in a forthcoming paper by the first author.