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}
$
\frac{{p + 3}}
{2}
directions. In
$
\frac{{p + 5}}
{2}
$
\frac{{p + 5}}
{2}
and
$
2\frac{{p - 1}}
{3}
$
2\frac{{p - 1}}
{3}
. The upper bound
obtained is one less than the smallest known example.
Mathematics Subject
Classification (2000): 51E15