发表机构
Georgia Institute of Technology（佐治亚理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对 Erdős 关于块设计横截与块交点数的猜想，构造反例证明当块大小为 $\Omega(\sqrt{n})$ 时，存在横截与某块交点达 $\Omega(\frac{\log n}{\log\log n})$，否定原猜想。
AI 中文摘要
块设计是一个集合族，其成员称为块，其中宇宙中每一对点恰好被一个块覆盖。Erdős 提出了以下问题：如果宇宙大小为 $n$ 且每个块大小为 $\Omega(\sqrt{n})$，那么是否存在一个横截与每个块相交于 $O(1)$ 个点？我们对此问题给出了否定答案。事实上，对于任意大的 $n$，我们构造了这样的块设计，其中每个横截与某个块相交于 $\Omega(\frac{\log n}{\log\log n})$ 个点。
英文摘要
A block design is a set family, whose members are called blocks, where every pair of points in the universe is covered by exactly one block. Erdős asked the following question: is it true that if the universe is of size $n$ and each block is of size $Ω(\sqrt{n})$, then there exists a transversal intersecting every block in $O(1)$ points? We provide a negative answer to this question. Indeed, for arbitrarily large $n$, we construct such a block designs where every transversal intersects some block in $Ω(\frac{\log n}{\log\log n})$ points.
Comments6 pages. Comments are welcome!
