Characterization of equality cases for relative and ordinary ordered Turán densities

Characterize the classes of ordered graphs \(F\) satisfying \(\mathfrak{o}(F)=\pi(F)\) and \(\mathfrak{o}(F)=\widetilde{\pi}(F)\), and determine whether any ordered graphs satisfy the strict intermediate inequality \(\widetilde{\pi}(F)<\mathfrak{o}(F)<\pi(F)\).

Background

The paper establishes the general inequality o(F)≤π~(F)\mathfrak{o}(F)\le \widetilde{\pi}(F) for ordered graphs and notes that ordered cliques attain equality. For ordered paths, Theorem 1.2 yields equality between the relative Turán density and the ordered Turán density. The authors ask for broader characterizations of both equality classes and whether a genuinely intermediate value can occur.

References

Problem 4.2. Characterise the class {F : (F) = ₸(F)}. For instance, all ordered cliques K, are in this class. Are there any other such graphs? Similarly, when F = Pk is a path, then Theorem 1.2 yields (F) = 27(F); we may thus ask for a characterisation of the class {F: o(F) =}7(F)}. Are there any graphs F satisfying 1 ₸(F) < (F) <₸(F)?

— Relative Turán densities of ordered graphs  (2501.06853 - Reiher et al., 12 Jan 2025) in Problem 4.2, Section 4 (Concluding Remarks)