Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Thursday, October 05, 2017, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Dániel Korándi (EPFL)

On the Turán number of ordered forests

An ordered graph $H$ is a graph with a linear ordering on its vertex set. The corresponding Turán problem, first studied by Pach and Tardos, asks for the maximum number $\ex_ n^{1+\eps}$ for some positive $\eps=\eps(H)$ unless $H$ is a forest that has a bipartition $V_1\cup V_2$ such that $V_1$ totally precedes $V_2$ in the ordering. Making progress towards a conjecture of Pach and Tardos, we prove that $\ex_{\le}(n,H) =n^{1+o(1)}$ 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. For example, the class contains all forests with $|V_1|\le 3$. Our proof is based on a density-increment argument. Joint work with Gábor Tardos, István Tomon and Craig Weidert

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