论文标题
超图中不可避免的订单尺寸对 - 正迫使密度
Unavoidable order-size pairs in hypergraphs -- positive forcing density
论文作者
论文摘要
ErdőS，Füredi，Rothschild和Sós开始了一类图表的研究，这些图形禁止在给定数字$ M $的顶点和数字$ f $边缘上的每个引起的子图。将其符号扩展到$ r $ - 绘图，我们写$（n，e）\ to_r（m，f）$，如果每个$ r $ graph $ g $ in $ n $ n $ dertices的每个$ e $ e $ edges都具有$ m $ to的诱导子级$ m $ dertices和$ f $ edges。一对$（m，f）$的\ emph {强迫密度}是$$σ_r（m，f）= \ left。 \ limsup \ limits_ {n \ to \ infty} \ frac {| \ { 。韦伯询问是否有一对$ r \ geq 3 $的积极强度，除了琐碎的$（m，0）$和$（m，\ binom {m} {m} {r}）$。回答她的问题，我们表明$（6,10）$是$ r = 3 $的一对，并猜想它是独特的这对。此外，我们发现一对具有阳性强度密度的必要条件，并支持该猜想。
Erdős, Füredi, Rothschild and Sós initiated a study of classes of graphs that forbid every induced subgraph on a given number $m$ of vertices and number $f$ of edges. Extending their notation to $r$-graphs, we write $(n,e) \to_r (m,f)$ if every $r$-graph $G$ on $n$ vertices with $e$ edges has an induced subgraph on $m$ vertices and $f$ edges. The \emph{forcing density} of a pair $(m,f)$ is $$ σ_r(m,f) =\left. \limsup\limits_{n \to \infty} \frac{|\{e : (n,e) \to_r (m,f)\}|}{\binom{n}{r}} \right. .$$ In the graph setting it is known that there are infinitely many pairs $(m, f)$ with positive forcing density. Weber asked if there is a pair of positive forcing density for $r\geq 3$ apart from the trivial ones $(m, 0)$ and $(m, \binom{m}{r})$. Answering her question, we show that $(6,10)$ is such a pair for $r=3$ and conjecture that it is the unique such pair. Further, we find necessary conditions for a pair to have positive forcing density, supporting this conjecture.
