Bounds on the Number of Compatible k-Simplices Matching the Orientation of the (k − 1)-Skeleton of a Simplex

We extend the notion of tournaments to orientation of the (k − 1)-skeleton of an (n − 1)-dimensional simplex. We ask for the maximal number of k-simplices whose boundary matches the orientation, extending the question on the upper bound of the number of directed 3-cycles of a tournament. In the general case we show polynomial upper and lower bounds. For the case k = 3 we improve the lower bound. Furthermore, this lower bound reaches the upper bound when n or n − 1 is a prime power congruent to 3 modulo 4.