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DIRECTED SIMPLICES IN HIGHER ORDER TOURNAMENTS

Published online by Cambridge University Press:  10 December 2009

Imre Leader*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, U.K. (email: [email protected])
Ta Sheng Tan
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, U.K. (email: [email protected])
*
For correspondence; e-mail: [email protected]
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Abstract

It is well known that a tournament (complete oriented graph) on n vertices has at most directed triangles, and that the constant 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.

Type
Research Article
Copyright
Copyright © University College London 2010

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