What is the shortest non-trivial logical deduction about overlapping circles? 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? 
 A: 
If three or more (but only finitely-many) circular regions in the plane are such that any three have a point in common, then all of them have a point in common. 

(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: 

If $d+1$ or more (but only finitely-many) convex subsets of $\mathbb{R}^d$ are such that any $d+1$ of them have a point in common, then all of them have a point in common. 

(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".
