Lagrangians are attained as uniform Turán densities
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910736215703552 |
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| author | King, Dylan Sales, Marcelo Schülke, Bjarne |
| author_facet | King, Dylan Sales, Marcelo Schülke, Bjarne |
| contents | The study of uniform Turán densities was initiated in the 1980s by Erdős and Sós. Given a $3$-graph $F$, the uniform Turán density of $F$, $π_{\therefore}(F)$, is defined as the infimum $d\in[0,1]$ such that every $3$-graph $H$ in which every linearly sized $S\subseteq V(H)$ induces at least $(d+o(1))\binom{\vert S\vert}{3}$ edges must contain a copy of $F$. Disproving Erdős's famous jumping conjecture, Frankl and Rödl showed that the set of Turán densities is not well-ordered. We prove an analogous result for the uniform Turán density, namely that the set $Π^{(3)}_{\therefore,\infty}=\{π_{\therefore}(\mathcal{F}) : \mathcal{F}\text{ a family of }3\text{-graphs} \}$ is not well-ordered. This is a consequence of a more general result, which in particular implies that for every Lagrangian $Λ$ of a $3$-graph and integer $1 \leq t \leq 6$ we have $\frac{t}{6}Λ\in Π^{(3)}_{\therefore,\infty}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_07297 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lagrangians are attained as uniform Turán densities King, Dylan Sales, Marcelo Schülke, Bjarne Combinatorics 05C65 The study of uniform Turán densities was initiated in the 1980s by Erdős and Sós. Given a $3$-graph $F$, the uniform Turán density of $F$, $π_{\therefore}(F)$, is defined as the infimum $d\in[0,1]$ such that every $3$-graph $H$ in which every linearly sized $S\subseteq V(H)$ induces at least $(d+o(1))\binom{\vert S\vert}{3}$ edges must contain a copy of $F$. Disproving Erdős's famous jumping conjecture, Frankl and Rödl showed that the set of Turán densities is not well-ordered. We prove an analogous result for the uniform Turán density, namely that the set $Π^{(3)}_{\therefore,\infty}=\{π_{\therefore}(\mathcal{F}) : \mathcal{F}\text{ a family of }3\text{-graphs} \}$ is not well-ordered. This is a consequence of a more general result, which in particular implies that for every Lagrangian $Λ$ of a $3$-graph and integer $1 \leq t \leq 6$ we have $\frac{t}{6}Λ\in Π^{(3)}_{\therefore,\infty}$. |
| title | Lagrangians are attained as uniform Turán densities |
| topic | Combinatorics 05C65 |
| url | https://arxiv.org/abs/2412.07297 |