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


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.

This is a preview of subscription content, access via your institution.


  1. [1]

    R. C. Baker, G. Harman and J. Pintz: The difference between consecutive primes. II., Proc. London Math. Soc. (3) 83 (2001), 532–562.

    MathSciNet  Article  Google Scholar 

  2. [2]

    D. M. Berman: On the number of 5-cycles in a tournament, in: Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), 101–108. Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Man., 1975.

  3. [3]

    U. Colombo: Sui circuiti nei grafi completi, Boll. Un. Mat. Ital. (3) 19 (1964), 153–170.

    MathSciNet  MATH  Google Scholar 

  4. [4]

    G. Davidoff, P. Sarnak and A. Valette: Elementary number theory, group theory, and Ramanujan graphs, London Mathematical Society Student Texts, 55. Cambridge University Press, Cambridge, 2003.

    MATH  Google Scholar 

  5. [5]

    D. Gale: On the number of faces of a convex polytope, Canad. J. Math. 16 (1964), 12–17.

    MathSciNet  Article  Google Scholar 

  6. [6]

    M. G. Kendall and B. B. Smith: On the method of paired comparisons, Biometrika 31 (1940) 324–345.

    MathSciNet  Article  Google Scholar 

  7. [7]

    N. Komarov and J. Mackey: On the number of 5-cycles in a tournament, J. Graph Theory 86 (2017), 341–356.

    MathSciNet  Article  Google Scholar 

  8. [8]

    A. Kotzig: Sur le nombre des 4-cycles dans un tournoi, Mat. Časopis Sloven. Akad. Vied 18 (1968), 247–254.

    MathSciNet  MATH  Google Scholar 

  9. [9]

    J. W. Moon: Topics on tournaments, Holt, Rinehart and Winston, New York-Montreal, Que.-London, 1968.

    MATH  Google Scholar 

  10. [10]

    J. R. Munres: Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984.

    Google Scholar 

  11. [11]

    S. V. Savchenko: On 5-cycles and 6-cycles in regular n-tournaments, J. Graph Theory 83 (2016), 44–77.

    MathSciNet  Article  Google Scholar 

  12. [12]

    S. V. Savchenko: On the number of 7-cycles in regular n-tournaments, Discrete Math. 340 (2017), 264–285.

    MathSciNet  Article  Google Scholar 

  13. [13]

    T. Szele: Kombinatorikai vizsgálatok az irányított teljes gráffal kapcsolatban, Mat. Fiz. Lapok 50 (1943) 223–256. For a German translation, see Kombinatorische Untersuchungen über gerichtete vollständige Graphen, Publ. Math. Debrecen 13 (1966), 145–168.

    MathSciNet  Google Scholar 

Download references


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).

Author information



Corresponding author

Correspondence to Richard Ehrenborg.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation

Mathematics Subject Classification (2010)

  • Primary 05C65
  • Secondary 05C20