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- W4296556710 abstract "The Tur'an density of an $r$-uniform hypergraph $H$, denoted $pi(H)$, is the limit of the maximum density of an $n$-vertex $r$-uniform hypergraph not containing a copy of $H$, as $n to infty$. Denote by $C_{ell}$ the $3$-uniform tight cycle on $ell$ vertices. Mubayi and Rodl gave an ``iterated blow-up'' construction showing that the Tur'an density of $C_5$ is at least $2sqrt{3} - 3 approx 0.464$, and this bound is conjectured to be tight. Their construction also does not contain $C_{ell}$ for larger $ell$ not divisible by $3$, which suggests that it might be the extremal construction for these hypergraphs as well. Here, we determine the Tur'an density of $C_{ell}$ for all large $ell$ not divisible by $3$, showing that indeed $pi(C_{ell}) = 2sqrt{3} - 3$. To our knowledge, this is the first example of a Tur'an density being determined where the extremal construction is an iterated blow-up construction. A key component in our proof, which may be of independent interest, is a $3$-uniform analogue of the statement ``a graph is bipartite if and only if it does not contain an odd cycle''." @default.
- W4296556710 created "2022-09-21" @default.
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- W4296556710 date "2022-09-16" @default.
- W4296556710 modified "2023-09-29" @default.
- W4296556710 title "The Tur'an density of tight cycles in three-uniform hypergraphs" @default.
- W4296556710 doi "https://doi.org/10.48550/arxiv.2209.08134" @default.
- W4296556710 hasPublicationYear "2022" @default.
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