Restricted Point Configurations with Many Collinear {k}-Tuplets

AbstractGiven k ≥ 3, denote by t′k(N) the largest integer for which there is a set of N points in the plane, no k+1 of them on a line such that there are t′k(N) lines, each containing exactly k of the points. Erdős (1962) raised the problem of estimating the order of magnitude of t′k(N). We prove that $$t'_k (N) \geqslant \left\{ \begin{gathered} c'_k N^{\log (k + 4)/\log (k)} if 4 \leqslant k \leqslant 17, \hfill \\ c'_k N^{1 + 1/(k - 3.59)} if k \geqslant 18, \hfill \\ \end{gathered} \right.$$ improving a previous bound of Grunbaum for all k ≥ 5. The proof for k ≥ 18 uses an argument of Brass with his permission.