On High-Dimensional Acyclic Tournaments

Nati Linial; Avraham Morgenstern

doi:10.1007/s00454-013-9543-8

On High-Dimensional Acyclic Tournaments

Nati Linial, Avraham Morgenstern^*

^*Corresponding author for this work

The Rachel and Selim Benin School of Engineering and Computer Science

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

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 aswell the inter-relations among various other notions of acyclicity in high-dimensional tournaments. These include combinatorial, geometric and topological concepts.

Original language	English
Pages (from-to)	1085-1100
Number of pages	16
Journal	Discrete and Computational Geometry
Volume	50
Issue number	4
DOIs	https://doi.org/10.1007/s00454-013-9543-8
State	Published - Dec 2013

Bibliographical note

Funding Information:
We were not sure for a while which of the many notions of acyclicity would be of greatest interest to study. We are grateful to Roy Meshulam for helping us take the (hopefully) right decision. Research supported in part by the Israel Science Foundation and by a USA–Israel BSF Grant

Keywords

Acyclic
Enumeration
High-dimensional
Hyper-plane arrangements
Tournament

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10.1007/s00454-013-9543-8

Cite this

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N2 - 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(log1/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 aswell the inter-relations among various other notions of acyclicity in high-dimensional tournaments. These include combinatorial, geometric and topological concepts.

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KW - Enumeration

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KW - Hyper-plane arrangements

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On High-Dimensional Acyclic Tournaments

Abstract

Bibliographical note

Keywords

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