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Ordered and Convex Geometric Trees with Linear Extremal Function

  • Zoltán Füredi
  • Alexandr KostochkaEmail author
  • Dhruv Mubayi
  • Jacques Verstraëte
Branko Grünbaum Memorial Issue
  • 5 Downloads

Abstract

The extremal functions \(\mathrm{{ex}}_{\rightarrow }(n,F)\) and \(\mathrm{{ex}}_{\circlearrowright }(n,F)\) for ordered and convex geometric acyclic graphs F have been extensively investigated by a number of researchers. Basic questions are to determine when \(\mathrm{{ex}}_{\rightarrow }(n,F)\) and \(\mathrm{{ex}}_{\circlearrowright }(n,F)\) are linear in n, the latter posed by Brass–Károlyi–Valtr in 2003. In this paper, we answer both these questions for every tree F. We give a forbidden subgraph characterization for a family \({\mathcal {T}}\) of ordered trees with k edges, and show that \(\mathrm{{ex}}_{\rightarrow }(n,T) = (k - 1)n - {k \atopwithdelims ()2}\) for all \(n \ge k + 1\) when \(T \in {{\mathcal {T}}}\) and \(\mathrm{{ex}}_{\rightarrow }(n,T) = \Omega (n\log n)\) for \(T \not \in {{\mathcal {T}}}\). We also describe the family \({{\mathcal {T}}}'\) of the convex geometric trees with linear Turán number and show that for every convex geometric tree \(F\notin {{\mathcal {T}}}'\), \(\mathrm{{ex}}_{\circlearrowright }(n,F)= \Omega (n\log \log n)\).

Keywords

Convex geometric graphs Ordered graphs Turán number 

Mathematics Subject Classification

52C30 05C35 05C05 

Notes

Acknowledgements

This research was partly conducted during AIM SQuaRes (Structured Quartet Research Ensembles) workshops, and we gratefully acknowledge the support of AIM. We also thank the referees for their comments.

References

  1. 1.
    Bienstock, D., Győri, E.: An extremal problem on sparse 0–1 matrices. SIAM J. Discrete Math. 4(1), 17–27 (1991)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brass, P., Károlyi, Gy., Valtr, P.: A Turán-type extremal theory of convex geometric graphs. In: Aronov, B., et al. (eds.) Discrete and Computational Geometry: The Goodman–Pollack Festscrift. Algorithms and Combinatorics, vol. 25, pp. 275–300. Springer, Berlin (2003)Google Scholar
  3. 3.
    Füredi, Z., Hajnal, P.: Davenport–Schinzel theory of matrices. Discrete Math. 103(3), 233–251 (1992)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Klazar, M.: The Füredi–Hajnal conjecture implies the Stanley–Wilf conjecture. In: Krob, D., Mikhalev, A.A., Mikhalev, A.V. (eds.) Formal Power Series and Algebraic Combinatorics, pp. 250–255. Springer, Berlin (2000)CrossRefGoogle Scholar
  5. 5.
    Korándi, D., Tardos, G., Tomon, I., Weidert, C.: On the Turán number of ordered forests. arXiv:1711.07723 (2017)
  6. 6.
    Kupitz, Y.S., Perles, M.A.: Extremal theory for convex matchings in convex geometric graphs. Discrete Comput. Geom. 15(2), 195–220 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Moser, W., Pach, J.: Recent developments in combinatorial geometry. In: Pach, J. (ed.) New Trends in Discrete and Computational Geometry. Algorithms and Combinatorics, vol. 10, pp. 281–302. Springer, New York (1993)Google Scholar
  8. 8.
    Marcus, A., Tardos, G.: Excluded permutation matrices and the Stanley–Wilf conjecture. J. Comb. Theory Ser. A 107(1), 153–160 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Pach, J., Tardos, G.: Forbidden patterns and unit distances. In: Computational Geometry (SCG’05), pp. 1–9. ACM, New York (2005)Google Scholar
  10. 10.
    Pettie, S.: Degrees of nonlinearity in forbidden 0–1 matrix problems. Discrete Math. 311(21), 2396–2410 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Tardos, G.: On 0–1 matrices and small excluded submatrices. J. Combin. Theory Ser. A 111(2), 266–288 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Tardos, G.: Construction of locally plane graphs with many edges. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 541–562. Springer, New York (2013)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary
  2. 2.University of Illinois at Urbana–ChampaignUrbanaUSA
  3. 3.Sobolev Institute of MathematicsNovosibirskRussia
  4. 4.Department of Mathematics, Statistics and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA
  5. 5.Department of MathematicsUniversity of California at San DiegoLa JollaUSA

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