# Ordered and convex geometric trees with linear extremal function

## TL;DR

This paper characterizes when extremal functions for ordered and convex geometric trees are linear in the number of vertices, providing forbidden subgraph criteria and precise bounds for various tree classes.

## Contribution

It offers a forbidden subgraph characterization for ordered trees with linear extremal functions and identifies convex geometric trees with linear Turán numbers, resolving key open questions.

## Key findings

- For ordered trees in family , extremal function is linear: (k-1)n - C(k,2).
- For ordered trees outside , extremal function grows at least as fast as n log n.
- Convex geometric trees outside a specific family have extremal functions at least (n) log log n.

## Abstract

The extremal functions $ex_{\rightarrow}(n,F)$ and $ex_{\cir}(n,F)$ for ordered and convex geometric acyclic graphs $F$ have been extensively investigated by a number of researchers. Basic questions are to determine when $ex_{\rightarrow}(n,F)$ and $ex_{\cir}(n,F)$ are linear in $n$, the latter posed by Bra\ss-K\'arolyi-Valtr in 2003. In this paper, we answer both these questions for every tree $F$. We give a forbidden subgraph characterization for a family $\cal T$ of ordered trees with $k$ edges, and show that $ex_{\rightarrow}(n,T) = (k - 1)n - {k \choose 2}$ for all $n \geq k + 1$ when $T \in {\cal T}$ and $ex_{\rightarrow}(n,T) = \Omega(n\log n)$ for $T \not\in {\cal T}$. We also describe the family of the convex geometric trees with linear Tur\' an number and show that for every convex geometric tree $F$ not in this family, $ex_{\cir}(n,F)= \Omega(n\log \log n)$.