A note on order-type homogeneous point sets

Let OTd(n) be the smallest integer N such that every N -element point sequence in R d in general position contains an order-type homogeneous subset of size n, where a set is order-type homogeneous if all (d + 1)-tuples from this set have the same orientation. It is known that a point sequence in R that is order-type homogeneous, forms the vertex set of a convex polytope that is combinatorially equivalent to a cyclic polytope in R. Two famous theorems of Erdős and Szekeres from 1935, imply that OT1(n) = Θ(n ) and OT2(n) = 2 . For d ≥ 3, we give new bounds for OTd(n). In particular: • We show that OT3(n) = 2 2 , answering a question of Eliáš and Matoušek. • For d ≥ 4, we show that OTd(n) = 2 . . 2O(n) , where the height of the tower is d.