Ordered and Convex Geometric Trees with Linear Extremal Function

Zoltán Füredi, Alexandr Kostochka, Dhruv Mubayi, Jacques Verstraëte

Research output: Article

Abstract

The extremal functions ex (n, F) and ex (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 ex (n, F) and ex (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 T of ordered trees with k edges, and show that ex→(n,T)=(k-1)n-(k2) for all n≥ k+ 1 when T∈ T and ex (n, T) = Ω (nlog n) for T∉ T. We also describe the family T of the convex geometric trees with linear Turán number and show that for every convex geometric tree F∉ T, ex (n, F) = Ω (nlog log n).

Original languageEnglish
JournalDiscrete and Computational Geometry
DOIs
Publication statusAccepted/In press - jan. 1 2019

Fingerprint

Extremal Function
Trees (mathematics)
Linear Function
Forbidden Subgraph
Ordered Trees
Graph in graph theory
Family

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Cite this

Ordered and Convex Geometric Trees with Linear Extremal Function. / Füredi, Zoltán; Kostochka, Alexandr; Mubayi, Dhruv; Verstraëte, Jacques.

In: Discrete and Computational Geometry, 01.01.2019.

Research output: Article

Füredi, Zoltán ; Kostochka, Alexandr ; Mubayi, Dhruv ; Verstraëte, Jacques. / Ordered and Convex Geometric Trees with Linear Extremal Function. In: Discrete and Computational Geometry. 2019.
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