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)>n1+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)=n1+o(1) holds for all such forests that are “degenerate” in a certain sense. This class includes every forest for which an n1+o(1) upper bound was previously known, as well as new examples. Our proof is based on a density-increment argument.
- Forbidden submatrix
- Ordered forest
- Turán problem
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics