论文标题
三均匀循环的turán密度
The Turán density of tight cycles in three-uniform hypergraphs
论文作者
论文摘要
$ r $均匀的超graph $ \ Mathcal {h} $的Turán密度，表示为$π（\ Mathcal {h}）$，是$ n $ n $ n $ vertex $ r $ r $ r $ runiform graph的最大密度的极限，不包含$ \ \ nathcal {h hh h} $ as $ n fty fty fty fty ftty beftty。 用$ \ mathcal {c} _ {\ ell} $表示$ 3 $ - 均匀的紧身周期，$ \ ell $ vertices。 Mubayi和Rödl给出了``迭代的爆炸''结构，表明$ \ Mathcal {C} _5 $的Turán密度至少为$ 2 \ sqrt {3} {3} - 3 \ 3 \ of 0.464 $，这一界限是紧密的。他们的构造还不包含$ \ Mathcal {C} _ {\ ell} $，对于较大的$ \ ell $不可用$ 3 $除外，这表明它可能也是这些超图的极端结构。在这里，我们确定$ \ MATHCAL {C} _ {\ ell} $的Turán密度，所有大$ \ ell $不可用$ 3 $除外，这确实表明确实$π（\ Mathcal {c} _ {\ ell} _ {\ ell}）= 2 \ sqrt {3} {3} - 3} - 3 $。据我们所知，这是确定Turán密度的第一个例子，而极端结构是迭代的爆破结构。 我们的证明中的一个关键组成部分可能是独立的，它是该语句的$ 3 $均匀的类似物：``当且仅当它不包含奇数循环时，图形是二分之一''。
The Turán density of an $r$-uniform hypergraph $\mathcal{H}$, denoted $π(\mathcal{H})$, is the limit of the maximum density of an $n$-vertex $r$-uniform hypergraph not containing a copy of $\mathcal{H}$, as $n \to \infty$. Denote by $\mathcal{C}_{\ell}$ the $3$-uniform tight cycle on $\ell$ vertices. Mubayi and Rödl gave an ``iterated blow-up'' construction showing that the Turán density of $\mathcal{C}_5$ is at least $2\sqrt{3} - 3 \approx 0.464$, and this bound is conjectured to be tight. Their construction also does not contain $\mathcal{C}_{\ell}$ 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 Turán density of $\mathcal{C}_{\ell}$ for all large $\ell$ not divisible by $3$, showing that indeed $π(\mathcal{C}_{\ell}) = 2\sqrt{3} - 3$. To our knowledge, this is the first example of a Turán 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''.
