AI 中文总结
研究平面直线排列的一般位置问题，通过构造直线集合改进相关定理界，得到Hadwiger - Debrunner数下界，还给出特定点集构造，证明Balogh和Solymosi超饱和引理在对数因子上最优。
AI 中文摘要
对于所有\(\delta>0\)以及无穷多个\(n\in\mathbb{N}\)，我们证明存在\(\mathbb{R}^2\)中\(n\)条直线的集合\(L\)，其中不存在相交的四元组，但对于每个满足\(|L'|\geq n^{\frac{4}{5}+\delta}\)的子集\(L'\subset L\)，存在\(L'\)中的三条直线有公共交点。这改进了Balogh和Solymosi一个定理的对偶形式的界。作为结果，我们得到了Hadwiger - Debrunner数\(HD_2(p,3)\)的改进下界。我们还给出了对于所有\(0\leq s\leq1\)以及任意大的\(n\in\mathbb{N}\)，基数为\(|S|\geq n^{3 - s}\)的点集\(S\subset[n]^3\)的构造，使得\(S\)包含\(O(n^{6 - 4s})\)个共线三元组。这表明Balogh和Solymosi的一个超饱和引理在对数因子上是最优的。
英文摘要
For all $δ>0$ and infinitely many $n \in \mathbb N$, we show that there exists a set $L$ of $n$ lines in $\mathbb R^2$ such that there are no intersecting quadruples, but for every subset $L' \subset L$ such that $|L'| \geq n^{\frac{4}{5}+δ}$, there exist three lines from $L'$ with a common point of intersection. This gives an improved bound for a dual form of a theorem of Balogh and Solymosi. As a consequence, we derive an improved lower bound for the Hadwiger-Debrunner number $HD_2(p,3)$. We also give, for all $0 \leq s \leq 1$ and arbitrarily large $n \in \mathbb N$, a construction of a point set $S \subset [n]^3$ with cardinality $|S|\geq n^{3-s}$, such that $S$ contains $O(n^{6-4s})$ collinear triples. This shows that a supersaturation lemma of Balogh and Solymosi is optimal, up to logarithmic factors.
