On the general position subset selection problem

Let f(n, l) be the maximum integer such that every set of n points in the plane with at most l collinear contains a subset of f(n, l) points with no three collinear. First we prove that if l 6 O( √ n) then f(n, l) > Ω( √ n ln l ). Second we prove that if l 6 O(n) then f(n, l) > Ω( √ n logl n), which implies all previously known lower bounds on f(n, l) and improves them when l is not fixed. A more general problem is to consider subsets with at most k collinear points in a point set with at most l collinear. We also prove analogous results in this setting.