Bounds on the Number of Compatible k-Simplices Matching the Orientation of the (k − 1)-Skeleton of a Simplex

Abstract

We extend the notion of tournaments to orientation of the (k − 1)-skeleton of an (n − 1)-dimensional simplex. We ask for the maximal number of k-simplices whose boundary matches the orientation, extending the question on the upper bound of the number of directed 3-cycles of a tournament. In the general case we show polynomial upper and lower bounds. For the case k = 3 we improve the lower bound. Furthermore, this lower bound reaches the upper bound when n or n − 1 is a prime power congruent to 3 modulo 4.

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Acknowledgments

The authors thank David Leep for the proof of Lemma 4.4 and Peter Sarnak for directing us to the cross ratio. We thank Fernando Xuancheng Shao for pointing us to results about “primes in short intervals”. We also thank Margaret Readdy and the two referees for their comments on an earlier version of this paper. This work was partially supported by a grant from the Simons Foundation (#429370 to Richard Ehrenborg).

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Correspondence to Richard Ehrenborg.

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Chandrasekha, K., Ehrenborg, R. Bounds on the Number of Compatible k-Simplices Matching the Orientation of the (k − 1)-Skeleton of a Simplex. Combinatorica 41, 209–236 (2021). https://doi.org/10.1007/s00493-020-4220-z

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Mathematics Subject Classification (2010)

  • Primary 05C65
  • Secondary 05C20