Directed Simplices In Higher Order Tournaments

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 $n$ vertices has at most ${1/4}\binom{n}{3}$ 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 $n$-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.