Caps with free pairs of points.

Journal of Geometry 85 (2006), 35-41.

We say that two points $x,y$ of a cap $C$ form a free pair of points if any plane containing $x$ and $y$ intersects $C$ in at most three points. For given $N$ and $q$, we denote by $m(N,q)$ the maximum number of points in a cap of ${\rm PG}(N,q)$ that contains at least one free pair of points. It is straightforward to prove that $m(N,q) \le (q^{N-1}+2q-3)/(q-1)$, and it is known that this bound is sharp for $q=2$ and all $N$. We use geometric constructions to prove that this bound is sharp for all $q$ when $N\le 4$. We briefly survey the motivation for constructions of caps with free pairs of points which comes from the area of statistical experimental design.