论文标题
超图周期的相对Turán数字
Relative Turán Numbers for Hypergraph Cycles
论文作者
论文摘要
对于$ r $ - 均匀的超graph $ h $和$ r $ r $均匀的超graphs $ \ mathcal {f} $的家族，相对的turán数字$ \ mathrm {ex}（h，\ nathcal {f}）$是$ \ mathcal {f nathcal {f nathcal {f} $ free free sum $ h $ h $ h $ h。在本文中，我们为某些HyperGraph Cycles $ \ Mathcal {F} $的家族（例如Berge Cycles and Berge Cycles and Loside Cycles of Berge Cycles and Cycles of Berge Cycles $ \ Mathrm {ex}（H，\ Mathcal {f}）$提供了下限。特别是，如果$ \ Mathcal {c} _ \ ell^3 $表示所有$ 3 $ -Sioncriber berge $ \ ell $ -cycles和$ h $的集合，是一个3-均均匀的超graph，最高度$Δ$，我们证明\ [\ sathrm {ex}（ex}（ex}） δ^{ - 3/4-o（1）} e（h），\] \ [\ mathrm {ex}（h，h，\ mathcal {c} _5^{3}）\geδ^{ - 3/4-o（h）3/4-o（1）}
For an $r$-uniform hypergraph $H$ and a family of $r$-uniform hypergraphs $\mathcal{F}$, the relative Turán number $\mathrm{ex}(H,\mathcal{F})$ is the maximum number of edges in an $\mathcal{F}$-free subgraph of $H$. In this paper we give lower bounds on $\mathrm{ex}(H,\mathcal{F})$ for certain families of hypergraph cycles $\mathcal{F}$ such as Berge cycles and loose cycles. In particular, if $\mathcal{C}_\ell^3$ denotes the set of all $3$-uniform Berge $\ell$-cycles and $H$ is a 3-uniform hypergraph with maximum degree $Δ$, we prove \[\mathrm{ex}(H,\mathcal{C}_4^{3})\ge Δ^{-3/4-o(1)}e(H),\] \[\mathrm{ex}(H,\mathcal{C}_5^{3})\ge Δ^{-3/4-o(1)}e(H),\] and these bounds are tight up to the $o(1)$ term.
