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It is well known that a tournament (complete oriented graph) on  n  vertices has at most1  4    (n3)directed triangles, and that the constant1  4is best possible. Motivated by some geometric considerations, our aim in this paper is to consider some “higher order” versions of this statement. For example, if we give each 3‐set from an  n ‐set a cyclic ordering, then what is the greatest number of “directed 4‐sets” we can have? We give an asymptotically best possible answer to this question, and give bounds in the general case when we orient each  d ‐set from an  n ‐set.

DIRECTED SIMPLICES IN HIGHER ORDER TOURNAMENTS