Before revision:  Given a set S of n slopes and an integer k\geq 1, is it possible to draw a complete graph on k vertices in the plane using only slopes from S? Equivalently, does there exist a set K of k points in general position such that the slope of every segment between two points of K is in S? We also present a polynomial algorithm for this question when n\leq 2k-c, conditionally on a conjecture of R.E. Jamison.
After revision:  Given a set S of n slopes and an integer k\geq 1, is it possible to draw a complete graph on k vertices in the plane using only slopes from S? Equivalently, does there exist a set K of k points in general position such that the slope of every segment between two points of K is in S? We also present a polynomial algorithm for this question when n\leq 2k-c, conditional on a conjecture of R.E. Jamison.