The 1913 Helly's theorem states that any family ${\cal K}$ of $n\geq d+1$ convex sets in ${\mathbb R}^d$ can be pierced by a single point if and only if any $d+1$ of ${\cal K}$'s elements can. In 2002 Alon, Kalai, Matoušek and Meshulam ruled out the possibility of similar criteria for the existence of lines crossing multiple convex sets in dimension $d\geq 3$ -- for any $k\geq 3$, they described arbitrary large families ${\cal K}$ of convex sets in ${\mathbb R}^3$ so that any $k$ elements of ${\cal K}$ can be crossed by a line yet no $k+4$ of them can. Let ${\cal K}$ be a family of $n$ pairwise intersecting convex sets in ${\mathbb R}^3$. We show that there exists a line crossing $Θ(n)$ elements of ${\cal K}$. This resolves the most extensively studied variant of a problem by Martínez, Roldán-Pensado and Rubin (Discrete Comput. Geom. 2020) which was highlighted by Bárány and Kalai (Bull. Amer. Math. Soc. 2021). Our result adds to the very few sufficient (and non-trivial) conditions that have been known for the existence of line transversals to large families of convex sets. Our argument is based on a Ramsey-type result of independent interest for families of pairwise intersecting convex sets in ${\mathbb R}^2$, and the structure of line arrangements in ${\mathbb R}^3$.

翻译：1913年的Helly定理指出，对于${\mathbb R}^d$中任意一族包含$n\geq d+1$个凸集的集合${\cal K}$，当且仅当${\cal K}$中任意$d+1$个元素能被单点刺穿时，整个族也能被单点刺穿。2002年，Alon、Kalai、Matoušek和Meshulam排除了在$d\geq 3$维空间中存在类似判据以保证直线能与多个凸集相交的可能性——对于任意$k\geq 3$，他们构造了${\mathbb R}^3$中任意大的凸集族${\cal K}$，使得${\cal K}$中任意$k$个元素都能被一条直线穿过，但任意$k+4$个元素却不能。设${\cal K}$为${\mathbb R}^3$中$n$个成对相交凸集构成的族。我们证明存在一条直线穿过${\cal K}$中$Θ(n)$个元素。这解决了Martínez、Roldán-Pensado和Rubin（Discrete Comput. Geom. 2020）所提出问题中研究最深入的变体，该问题曾被Bárány和Kalai（Bull. Amer. Math. Soc. 2021）重点强调。我们的结果为已知的极少数（且非平凡的）充分条件增添了新内容，这些条件保证了直线横截大族凸集的存在性。我们的证明基于一个具有独立意义的Ramsey型结论（针对${\mathbb R}^2$中成对相交凸集族）以及${\mathbb R}^3$中直线排列的结构特性。