### 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 language | English |
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Journal | Discrete and Computational Geometry |

DOIs | |

Publication status | Accepted/In press - jan. 1 2019 |

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### ASJC Scopus subject areas

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

### Cite this

*Discrete and Computational Geometry*. https://doi.org/10.1007/s00454-019-00149-z

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

Research output: Article

*Discrete and Computational Geometry*. https://doi.org/10.1007/s00454-019-00149-z

}

TY - JOUR

T1 - Ordered and Convex Geometric Trees with Linear Extremal Function

AU - Füredi, Zoltán

AU - Kostochka, Alexandr

AU - Mubayi, Dhruv

AU - Verstraëte, Jacques

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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).

AB - 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).

KW - Convex geometric graphs

KW - Ordered graphs

KW - Turán number

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UR - http://www.scopus.com/inward/citedby.url?scp=85074823463&partnerID=8YFLogxK

U2 - 10.1007/s00454-019-00149-z

DO - 10.1007/s00454-019-00149-z

M3 - Article

AN - SCOPUS:85074823463

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

ER -