Relative Turán densities for ordered graphs: all and nothing
Freddie Illingworth, Arjun Ranganathan, Leo Versteegen, Ella Williams
TL;DR
The paper advances the theory of relative Turán densities for ordered graphs by introducing a universally optimal host family $R(m,d)$, and proving that $\rho_{<}(F)=\lim_{d\to\infty}\lim_{m\to\infty} \rho_{<}(F,R(m,d))$ for every ordered graph $F$. It develops a binary, $\{0,1\}^d$-based framework and a notion of $(\alpha,C)$-rich graphs to reduce the embedding problem to regularity-based coverings, enabling precise control over edge distributions via random embeddings. A complete characterization of zero relative density is achieved: $\rho_{<}(F)=0$ if and only if $F$ contains no monotone path of length two; this is established by introducing the family $H_k$ and proving $\rho_{<}(H_k)=0$ for all $k$ through a detailed inductive embedding argument. Collectively, these results place the relative-ordered-Turán problem on a firm universal footing, connect it to rich-graph embeddings, and set a clear agenda for determining $\rho_{<}(F)$ for broader families of ordered graphs. The work also highlights open questions about density gaps, optimal host constructions, and the precise growth requirements for the host parameters.
Abstract
Reiher, Rödl, Sales, and Schacht initiated the study of relative Turán densities of ordered graphs and showed that it is more subtle and interesting than the unordered case. For an ordered graph $F$, its relative Turán density, $ρ_{<}(F)$, is the greatest $α$ such that every ordered graph $G$ has an $F$-free subgraph with at least $αe(G)$ edges. This paper contains two main results about relative Turán densities. First, we find a family of host graphs that is optimal for all $F$. Second, we characterise the ordered graphs with zero relative Turán density: precisely those with no monotone path of length two.