论文标题
统一超图的相对图片问题
Relative Turán Problems for Uniform Hypergraphs
论文作者
论文摘要
对于两个图$ f $和$ h $，相对Turán$ \ mathrm {ex}（h，f）$是$ f $ f $ f $ f $ free子图的最大边数。 foucaud，krivelevich和perarnau \ cite {fkp}和perarnau and reed \ cite {pr}研究了这些数量，这是$ h $的最大程度的函数。 在本文中，我们研究了均匀超图的概括。如果$ f $是一个完整的$ r $ -partite $ r $ r $均匀的超图，带有尺寸$ s_1，s_2，\ dots，s_r $的零件，每个$ s_ {i + 1} $相对于$ $ s_i $，则有足够大的大型对于任何$ r $ - 均匀的超图$ h $，具有最高度$δ$，\ [\ mathrm {ex}（h，h，f）\geδ^^Δ^{ - β-β-o（1）} \ cdot e（h）。 $ r $ -graph $ h $使得$ \ mathrm {ex}（h，f）= o（δ^{ - β}）\ cdot e（h）$。当$ h $是随机$ n $ -vertex $ r $ -graph $ h_ {n，p}^r $带Edge-probobility $ p $时，也会获得类似的紧缩结果，从而扩展了Balogh和Samotij \ cite {bs}的结果，以及Morris和Morris and Saxton \ Saxton \ cite {MS}。
For two graphs $F$ and $H$, the relative Turán number $\mathrm{ex}(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. Foucaud, Krivelevich, and Perarnau \cite{FKP} and Perarnau and Reed \cite{PR} studied these quantities as a function of the maximum degree of $H$. In this paper, we study a generalization for uniform hypergraphs. If $F$ is a complete $r$-partite $r$-uniform hypergraph with parts of sizes $s_1,s_2,\dots,s_r$ with each $s_{i + 1}$ sufficiently large relative to $s_i$, then with $1/β= \sum_{i = 2}^r \prod_{j = 1}^{i - 1} s_j$ we prove that for any $r$-uniform hypergraph $H$ with maximum degree $Δ$, \[\mathrm{ex}(H,F)\ge Δ^{-β- o(1)} \cdot e(H).\] This is tight as $Δ\rightarrow \infty$ up to the $o(1)$ term in the exponent, since we show there exists a $Δ$-regular $r$-graph $H$ such that $\mathrm{ex}(H,F)=O(Δ^{-β}) \cdot e(H)$. Similar tight results are obtained when $H$ is the random $n$-vertex $r$-graph $H_{n,p}^r$ with edge-probability $p$, extending results of Balogh and Samotij \cite{BS} and Morris and Saxton \cite{MS}.
