Turán Densities for Daisies and Hypercubes
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917836834734080 |
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| author | Ellis, David Ivan, Maria-Romina Leader, Imre |
| author_facet | Ellis, David Ivan, Maria-Romina Leader, Imre |
| contents | An $r$-daisy is an $r$-uniform hypergraph consisting of the six $r$-sets formed by taking the union of an $(r-2)$-set with each of the 2-sets of a disjoint 4-set. Bollobás, Leader and Malvenuto, and also Bukh, conjectured that the Turán density of the $r$-daisy tends to zero as $r \to \infty$. In this paper we disprove this conjecture.
Adapting our construction, we are also able to disprove a folklore conjecture about Turán densities of hypercubes. For fixed $d$ and large $n$, we show that the smallest set of vertices of the $n$-dimensional hypercube $Q_n$ that meets every copy of $Q_d$ has asymptotic density strictly below $1/(d+1)$, for all $d \geq 8$. In fact, we show that this asymptotic density is at most $c^d$, for some constant $c<1$. As a consequence, we obtain similar bounds for the edge-Turán densities of hypercubes. We also answer some related questions of Johnson and Talbot, and disprove a conjecture made by Bukh and by Griggs and Lu on poset densities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_16289 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Turán Densities for Daisies and Hypercubes Ellis, David Ivan, Maria-Romina Leader, Imre Combinatorics 05C65 An $r$-daisy is an $r$-uniform hypergraph consisting of the six $r$-sets formed by taking the union of an $(r-2)$-set with each of the 2-sets of a disjoint 4-set. Bollobás, Leader and Malvenuto, and also Bukh, conjectured that the Turán density of the $r$-daisy tends to zero as $r \to \infty$. In this paper we disprove this conjecture. Adapting our construction, we are also able to disprove a folklore conjecture about Turán densities of hypercubes. For fixed $d$ and large $n$, we show that the smallest set of vertices of the $n$-dimensional hypercube $Q_n$ that meets every copy of $Q_d$ has asymptotic density strictly below $1/(d+1)$, for all $d \geq 8$. In fact, we show that this asymptotic density is at most $c^d$, for some constant $c<1$. As a consequence, we obtain similar bounds for the edge-Turán densities of hypercubes. We also answer some related questions of Johnson and Talbot, and disprove a conjecture made by Bukh and by Griggs and Lu on poset densities. |
| title | Turán Densities for Daisies and Hypercubes |
| topic | Combinatorics 05C65 |
| url | https://arxiv.org/abs/2401.16289 |