Bounds assuring subsets in convex position

For n≥the Ramsey number N(q, q;d+1), every general n-ad in d-space has a comonotone q-sub-ad; for d=2 “comonotone” becomes convex. Likewise, every 2N(q, q;3)-set of half-planes without parallelism or concurrence of bounding lines has a q-subset that consists either of the defining half-planes of an unbounded convex polyhedral set or of the complementary half-planes.