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For every $k>3$, we give a construction of planar point sets with manycollinear $k$-tuples and no collinear $(k+1)$-tuples. We show that there are$n_0=n_0(k)$ and $c=c(k)$ such that if $n\geq n_0$, then there exists a set of$n$ points in the plane that does not contain $k+1$ points on a line, but itcontains at least $n^{2-\frac{c}{\sqrt{\log n}}}$ collinear $k$-tuples ofpoints. Thus, we significantly improve the previously best known lower boundfor the largest number of collinear $k$-tuples in such a set, and getreasonably close to the trivial upper bound $O(n^2)$.

Many collinear k-tuples with no k+1 collinear points