For line segments in a 1D infinite space the shortest non-trivial statement I can think of is:
"For line segments $A$, $B$ and $C$. If there are regions where (at least) $A$ and $B$ overlap and regions where $B$ and $C$ overlap and regions where $A$ and $C$ overlap there must be a region where $A$, $B$ and $C$ overlap."
Now in a 2D infinite plane with a number of filled circles $A,B,C,D...$ , you are told some incomplete information about if there are areas where the circles overlap and which circles these are. What it the simplest non-trivial logical deduction that one could make from this information about overlaps in the same sort of way as was done with line segments?
(The case of three regions is, of course, tautological, but including it makes for the most-complete statement of the result.) This, and OP's segment example, are special cases of Helly's Theorem, which can be expressed as:
(Again, the case of $d+1$ subsets is tautological.) As the Wikipedia article notes, the version of the theorem for infinitely-many regions requires the regions to be compact as well as convex.
It's worth noting that the topology of $\mathbb{R}^d$ is important here. If OP's example were not about segments on the line but arcs on a circle, pairwise intersections do not imply a common intersection. Nor does the statement I mentioned if "circular regions in the plane" is replaced by "caps on the sphere".