Ordered and Convex Geometric Trees with Linear Extremal Function

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


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 \(F\notin {{\mathcal {T}}}'\), \(\mathrm{{ex}}_{\circlearrowright }(n,F)= \Omega (n\log \log n)\).


Convex geometric graphs Ordered graphs Turán number 

Mathematics Subject Classification

52C30 05C35 05C05 



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.


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© 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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