KöMaL 2013 A596

Find all integers $k \ge$3 with the property that for every infinite sequence $P_1$,$P_2$,... of points in the plane such that no three of them are collinear, there exists a sequence $1\le i_1<i_2<\ldots<i_k$ of integers for which the points $P_{i_1},P_{i_2},\ldots,P_{i_k}$, in that order, form a convex $k$-gon.