Relative Tur\'an densities for ordered graphs: all and nothing

Reiher, R\"odl, Sales, and Schacht initiated the study of relative Tur\'an densities of ordered graphs and showed that it is more subtle and interesting than the unordered case. For an ordered graph $F$, its relative Tur\'an density, $\rho_{<}(F)$, is the greatest $\alpha$ such that every ordered graph $G$ has an $F$-free subgraph with at least $\alpha e(G)$ edges. This paper contains two main results about relative Tur\'an densities. First, we find a family of host graphs that is optimal for all $F$. Second, we characterise the ordered graphs with zero relative Tur\'an density: precisely those with no monotone path of length two.