On the Turán Number of Ordered Forests

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.