On the Tur\'an number of ordered forests

TL;DR

This paper investigates the maximum edges in ordered graphs avoiding certain forests, proving that for a broad class of these forests, this maximum grows nearly linearly with the number of vertices, advancing understanding of ordered Turán problems.

Contribution

The authors prove that for a large class of degenerate ordered forests, the Turán number grows as n^{1+o(1)}, extending previous bounds and introducing a density-increment proof technique.

Findings

Established that ex_<(n,H) = n^{1+o(1)} for all degenerate forests with specific properties.

Extended known upper bounds to new classes of ordered forests.

Used a density-increment argument to prove the main results.

Abstract

An ordered graph $H$ is a simple graph with a linear order on its vertex set. The corresponding Tur\'an problem, first studied by Pach and Tardos, asks for the maximum number $\text{ex}_<(n,H)$ of edges in an ordered graph on $n$ vertices that does not contain $H$ as an ordered subgraph. It is known that $\text{ex}_<(n,H) > n^{1+\varepsilon}$ for some positive $\varepsilon=\varepsilon(H)$ unless $H$ is a forest that has a proper 2-coloring with one color class totally preceding the other one. Making progress towards a conjecture of Pach and Tardos, we prove that $\text{ex}_<(n,H) =n^{1+o(1)}$ holds for all such forests that are "degenerate" in a certain sense. This class includes every forest for which an $n^{1+o(1)}$ upper bound was previously known, as well as new examples. Our proof is based on a density-increment argument.