Special Point Sets in Finite Projective Planes

We consider the following three problems: 1. Let U be a q-subset of GF(q 2) with the properties 0; 1 2 U and u ? v is a square for all u; v 2 U. Does it follow that U consists of the elements of the subbeld GF(q)? Here q is odd. 2. Let f : GF(q) ! GF(q) be any function, and let be the set of diierence quotients (directions, slopes). What are the possibilities for jD f j? 3. Let B be a subset of PG(2; q), the Desarguesian projective plane of order q, such that every line contains at least one point of B. What are the possibilities for jBj? The third problem is the oldest of the three. The subset B is called a blocking set. To make the problem interesting we restrict ourselves to minimal blocking sets, that is blocking sets not containing a proper subset that is still a blocking set. The smallest possible blocking set is always a line. The most interesting problem is the next possible size. Essentially the problem is due to Richardson 15] who considered the plane of order 3, although already in 11] it is mentioned that the only minimal blocking sets in the Fano plane PG(2; 2) are the lines. In PG(2; 3) the next possible size is 6. The problem was made popular by di Paola 12] who determined the next possible size in the planes