On a Generalization of Rédei’s Theorem

In 1970 Rédei and Megyesi proved that a set of p points in AG(2,p), p prime, is a line, or it determines at least $$ \frac{{p + 3}} {2} $$ directions. In ’81 Lovász and Schrijver characterized the case of equality. Here we prove that the number of determined directions cannot be between $$ \frac{{p + 5}} {2} $$ and $$ 2\frac{{p - 1}} {3} $$. The upper bound obtained is one less than the smallest known example.