Theorem 8.6.1 — pairwise intersecting d-intervals have a transversal of size 2d²

Let $d \ge 1$ and let $\mathcal J$ be a finite family of $d$-intervals (unions of $d$ closed intervals on the real line) such that $J_1 \cap J_2 \ne \emptyset$ for every $J_1, J_2 \in \mathcal J$. Then $\mathcal J$ has a transversal of size $2d^2$: there is a set $X \subset \mathbb R$ with

For $d = 1$ this is the one-dimensional Helly theorem up to a constant (pairwise intersecting intervals even have a common point); for $d \ge 2$ no common point need exist, and the theorem shows that a number of points depending only on $d$ always suffices. The bound $2d^2$ is Alon's (1998); Kaiser (1997) obtained $d^2$ with topological methods.

Formalization Note $X$ is a Finset ℝ of cardinality at most $2d^2$ ("there exist $2d^2$ points", which may coincide). The empty family is allowed; it is covered by $X = \emptyset$.