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arXiv · 1302.1684

On high-dimensional acyclic tournaments

Nati Linial·Avraham Morgenstern
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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 $Ω(\log^{1/d}n)$ 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.

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Nati Linial, Avraham Morgenstern. 2013-07-11. On high-dimensional acyclic tournaments. https://doi.org/10.1007/s00454-013-9543-8

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