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?

  • $\begingroup$ If you really just want the "simplest" deduction then this is surely opinion-based. $\endgroup$ – Eric Wofsey Nov 19 '18 at 21:56
  • $\begingroup$ Well OK, by simplest I mean fewest circles and shortest statement. $\endgroup$ – zooby Nov 19 '18 at 21:57
  • $\begingroup$ Same as the line segments. If A and B intersect, B and C intersect, C and A intersect, then there is a region (possibly a point) where all 3 intersect $\endgroup$ – NazimJ Nov 19 '18 at 22:03
  • $\begingroup$ @NazimJ Not true! You could have a hole in the middle! $\endgroup$ – zooby Nov 19 '18 at 22:08

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".

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    $\begingroup$ Thanks. That answers the question. $\endgroup$ – zooby Nov 20 '18 at 0:41
  • $\begingroup$ BTW Is there a version of Helly's theoream where the N-spheres are on a the surface of an (N+1)-sphere? $\endgroup$ – zooby Nov 21 '18 at 13:25
  • $\begingroup$ A web search for "Helly theorem on sphere" turns up a number of references to "Helly-type" results. $\endgroup$ – Blue Nov 21 '18 at 17:09

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