On Relative Ordered Turán Density

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