On Relative Ordered Turán Density

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