On the generalized Erdös-Szekeres Conjecture — a new upper bound

We prove the following result: For every two natural numbers n and q, n ⩾ q + 2, there is a natural number E(n, q) satisfying the following: 1. (1) Let S be any set of points in the plane, no three on a line. If |S| ⩾ E(n, q), then there exists a convex n-gon whose points belong to S, for which the number of points of S in its interior is 0 (mod q). 2. (2) For fixed q, E(n,q) ⩽ 2c(q)·n, c(q) is a constant depends on q only. Part (1) was proved by Bialostocki et al. [2] and our proof is aimed to simplify the original proof. The proof of Part (2) is completely new and reduces the huge upper bound of [2] (a super-exponential bound) to an exponential upper bound.