Avoiding Secants of Given Size in Finite Projective Planes

ABSTRACT Let be a prime power and be a natural number. What are the possible cardinalities of point sets in a projective plane of order , which do not intersect any line at exactly points? This problem and its variants have been investigated before, in relation with blocking sets, untouchable sets or sets of even type, among others. In this article, we show a series of results which point out the existence of all or almost all possible values for , provided that is not close to the extremal values 0 or . Moreover, using polynomial techniques we show that for every prescribed list of numbers , there exists a point set with the following property: holds for the th line , .