On high-dimensional acyclic tournaments

TL;DR

This paper explores high-dimensional analogs of acyclic tournaments, providing bounds on their quantities, identifying large acyclic subtournaments, and connecting these concepts to hypergraph Ramsey numbers and other notions of acyclicity.

Contribution

It introduces bounds and properties of high-dimensional acyclic tournaments, extending classical results and establishing links to hypergraph Ramsey theory.

Findings

Bounds on the number of high-dimensional acyclic tournaments

Existence of large acyclic subtournaments of size (( ))

Connection between acyclic high-dimensional tournaments and hypergraph Ramsey numbers

Abstract

We study a high-dimensional analog for the notion of an acyclic (aka transitive) tournament. We give upper and lower bounds on the number of $d$-dimensional $n$-vertex acyclic tournaments. In addition, we prove that every $n$-vertex $d$-dimensional tournament contains an acyclic subtournament of $\Omega(\log^{1/d}n)$ vertices and the bound is tight. This statement for tournaments (i.e., the case $d=1$) is a well-known fact. We indicate a connection between acyclic high-dimensional tournaments and Ramsey numbers of hypergraphs. We investigate as well the inter-relations among various other notions of acyclicity in high-dimensional to tournaments. These include combinatorial, geometric and topological concepts.