On Relative Ordered Turán Density
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909816351358976 |
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| author | King, Dylan Lidický, Bernard Ouyang, Minghui Pfender, Florian Wang, Runze Xiang, Zimu |
| author_facet | King, Dylan Lidický, Bernard Ouyang, Minghui Pfender, Florian Wang, Runze Xiang, Zimu |
| contents | 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 $0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05515 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Relative Ordered Turán Density King, Dylan Lidický, Bernard Ouyang, Minghui Pfender, Florian Wang, Runze Xiang, Zimu Combinatorics 05C35 (primary) 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 $0$. |
| title | On Relative Ordered Turán Density |
| topic | Combinatorics 05C35 (primary) |
| url | https://arxiv.org/abs/2508.05515 |