On Relative Ordered Tur\'an Density

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 $0$.