论文标题
没有弱三角洲系统的家庭的上限
Upper Bounds For Families Without Weak Delta-Systems
论文作者
论文摘要
对于$ k \ geq3 $，据说$ k $ sets的集合形成\ emph {弱$δ$ - 系统}，如果该集合中的任何两个集合的交集具有相同的大小。 erdős和szemerédi询问了$ \ {1，\ dots，n \} $的最大家族$ \ Mathcal {f} $的大小，其中不包含弱$Δ$ - 系统。在本说明中，我们改进了作者的最佳上限和Arxiv：1606.09575的最佳上限，并证明\ [| \ Mathcal {f} | \ leq \ left（\ frac {2} {3} {3}θ（3}θ（c）+o（c）+o（1）+o（1）\ right）特别是，这表明\ [| \ Mathcal {f} | \ leq（1.8367 \ dots+o（1））^{n}。 \]
For $k\geq3$, a collection of $k$ sets is said to form a \emph{weak $Δ$-system} if the intersection of any two sets from the collection has the same size. Erdős and Szemerédi asked about the size of the largest family $\mathcal{F}$ of subsets of $\{1,\dots,n\}$ that does not contain a weak $Δ$-system. In this note we improve upon the best upper bound of the author and Sawin from arXiv:1606.09575 and show that \[ |\mathcal{F}|\leq\left(\frac{2}{3}Θ(C)+o(1)\right)^{n} \] where $Θ(C)$ is the capset capacity. In particular, this shows that \[ |\mathcal{F}|\leq(1.8367\dots+o(1))^{n}. \]
