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Abstract  LetOT  d    (  n  )be the smallest integer   N   such that every   N  ‐element point sequence inR   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 inR   d    that is order‐type homogeneous, forms the vertex set of a convex polytope that is combinatorially equivalent to a cyclic polytope inR   d   . Two famous theorems of Erdős and Szekeres from 1935 imply thatOT  1    (  n  )   =  Θ   (    n   2   )andOT  2    (  n  )   =    2    Θ  (  n  ). For    d  ≥  3   , we give new bounds forOT  d    (  n  )    . In particular, we show thatOT  3    (  n  )   =    22    Θ  (  n  ), answering a question of Eliáš and Matoušek, and, for    d  ≥  4   , we show thatOT  d    (  n  )is bounded above by an exponential tower of height   d   with    O  (  n  )    in the topmost exponent.

A NOTE ON ORDER‐TYPE HOMOGENEOUS POINT SETS