Directed Simplices In Higher Order Tournaments
Imre Leader, Ta Sheng Tan

TL;DR
This paper explores higher-order analogs of directed triangles in tournaments, establishing asymptotically optimal bounds for the maximum number of directed 4-sets and general orientations of d-sets, motivated by geometric considerations.
Contribution
It introduces and analyzes higher-order versions of directed triangles in tournaments, providing asymptotically tight bounds for the maximum number of directed 4-sets and general d-set orientations.
Findings
Asymptotically optimal bounds for directed 4-sets in 3-uniform orientations.
Bounds for orientations of d-sets in n-sets.
Extension of classical tournament triangle results to higher dimensions.
Abstract
It is well known that a tournament (complete oriented graph) on vertices has at most directed triangles, and that the constant 1/4 is 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 -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 -set from an -set.
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