Separating hypergraph Tur\'an densities
Erd\H{o}s asked whether, similarly as for graphs, the Tur\'an density of $K_4^{(3)}$ is the same as that of $K_5^{(3)-}$. This was disproved by Markstr\"om using a proof that relied on computer-aided flag algebra calculations. Here we give a flag-algebra-free proof that for all $k\geq 3$,...
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Main Authors | , , , , |
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Format | Journal Article |
Language | English |
Published |
11.10.2024
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Subjects | |
Online Access | Get full text |
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Summary: | Erd\H{o}s asked whether, similarly as for graphs, the Tur\'an density of
$K_4^{(3)}$ is the same as that of $K_5^{(3)-}$. This was disproved by
Markstr\"om using a proof that relied on computer-aided flag algebra
calculations. Here we give a flag-algebra-free proof that for all $k\geq 3$,
$\pi(K_{k+1}^{(k)})<\pi(K_{k+2}^{(k)-})$. We further illustrate our methods by
separating the Tur\'an densities of other hypergraphs without relying on
explicit bounds. For instance, we show that
$\pi(K_{\ell}^{(k)})<\pi(K_{\ell+1}^{(k)})$, for all $\ell> k\geq3$. |
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DOI: | 10.48550/arxiv.2410.08921 |