Theorem (Erdos-Szekeres 1935) Vm≥3，N(m)such that any set of n≥W(m) points in the plane,no three on a line,contains m points that are the vertices of a convex m-gon. Polygon: convex concavePolygon: convex concave ∀ m ≥ 3, ∃ N(m) such that any set of n ≥ N(m) points in the plane, no three on a line, contains m points that are the vertices of a convex m-gon. ∀ m ≥ 3, ∃ N(m) such that any set of n ≥ N(m) points in the plane, no three on a line, contains m points that are the vertices of a convex m-gon. Theorem (Erdős-Szekeres 1935)