### 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 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 ex_{<}(n,H)>n^{1+ε} for some positive ε=ε(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 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.

Original language | English |
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Pages (from-to) | 32-43 |

Number of pages | 12 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 165 |

DOIs | |

Publication status | Published - Jul 1 2019 |

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### Keywords

- Forbidden submatrix
- Ordered forest
- Turán problem

### ASJC Scopus subject areas

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

### Cite this

*Journal of Combinatorial Theory. Series A*,

*165*, 32-43. https://doi.org/10.1016/j.jcta.2019.01.006