On Relative Ordered Turán Density
Dylan King, Bernard Lidický, Minghui Ouyang, Florian Pfender, Runze Wang, Zimu Xiang
TL;DR
This paper studies the relative Turán density $\varrho(F)$ for ordered graphs, uncovering that $\varrho(F)$ can strictly lie between $\frac{1}{2}\vec{\pi}(F)$ and $\vec{\pi}(F)$ for certain families $F$, thereby answering RRSS25's question in the affirmative. It introduces the family $Q_{a,b}$ with $\vec{\pi}(Q_{a,b})=(a+b-1)/(a+b)$ and provides upper bounds $\varrho(Q_{a,b})\le a/(a+1)$, while showing $\varrho(Q_{a,b})\ge 1/2$ when $a$ is even, demonstrating a concrete gap. The paper also shows that two specific three-edge ordered matchings have $\varrho=0$ by two independent methods: a recursive layered construction $H_d$ and a quasi-random interval model $G(n,\varepsilon)$, with a unifying Local Extension Argument establishing $\varrho(\widetilde{F})=\varrho(F)$ and implying zero density in broader families $\{M_k\}$. Together, these results reveal nuanced behavior of relative density in the ordered setting and provide versatile techniques (blow-up invariance, quasi-random constructions, and local extensions) for analyzing $F$-free subgraphs.
Abstract
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 $0$.