On Lines Crossing Pairwise Intersecting Convex Sets in Three Dimensions

The 1913 Helly's theorem states that any family K of n≥d+1 convex sets in R^d can be pierced by a single point if and only if any d+1 of 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≥3 -- for any k≥3, they described arbitrary large families K of convex sets in R^3 so that any k elements of K can be crossed by a line yet no k+4 of them can. Let K be a family of n pairwise intersecting convex sets in R^3. We show that there exists a line crossing Θ(n) elements of 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 R^2, and the structure of line arrangements in R^3.