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arXiv · 1711.07723

On the Turán number of ordered forests

Dániel Korándi·Gábor Tardos·István Tomon·Craig Weidert
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Abstract

An ordered graph $H$ is a simple graph with a linear order on its vertex set. The corresponding Turán 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.

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Dániel Korándi, Gábor Tardos, István Tomon, Craig Weidert. 2017-11-21. On the Turán number of ordered forests. https://arxiv.org/abs/1711.07723

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