On Relative Ordered Tur\'an Density

TL;DR

This paper investigates the relative Turán density of ordered graphs, providing examples where it differs from known bounds and establishing that many ordered matchings have zero relative Turán density.

Contribution

The authors identify a family of ordered graphs with relative Turán density strictly between the known bounds, answering an open question and expanding understanding of Turán densities in ordered graphs.

Findings

Existence of ordered graphs with $ ho(F)$ strictly between $rac{oldsymbol{ ilde{ ho}(F)}}{2}$ and $oldsymbol{ ilde{ ho}(F)}$

Many ordered matchings have relative Turán density equal to 0

Provided explicit examples of such graphs and matchings

Abstract

For an ordered graph $F$, denote the Tur\'an density by $\vec{\pi}(F)$. The relative Tur\'an density, denoted by $\rho(F)$, is the supremum over $\alpha \in [0,1]$ such that every ordered graph $G$ contains an $F$-free subgraph $G'$ with $e(G') \geq \alpha e(G)$. Reiher, R\"odl, Sales and Schacht showed that $\rho(P) = \vec{\pi}(P)/2$ and $\rho(K) = \vec{\pi}(K)$ for any ascending path $P$ or clique $K$. They asked if there are any ordered graphs $F$ with $\vec{\pi}(F)/2 < \rho(F) < \vec{\pi}(F)$. We answer this question in the affirmative by describing a family of such $F$. We also show that the relative Tur\'an densities of a large family of ordered matchings (including $\{\{1,6\}, \{2,3\}, \{4,5\}\}$ and $\{\{1,3\}, \{2,5\}, \{4,6\}\}$) are $0$.