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I have one question related to differential geometry. Initilally, I am giving some background and my question is after that.

Helly's Theorem Let C be a finite family of convex sets in $R^n$ such that, for $k ≤ n + 1$, any $k$ members of C have a nonempty intersection. Then the intersection of all members of C is nonempty.

As an application of Helly's therore we have:

Corollary : Given $s (s > 0)$ points in the plane such that every three of them are contained in a disk of radius $1$. Prove that all $s$ points are contained in a disk of radius $1$.

Proof Consider the set C of unit disks with centers at the points from a given set. Since every three of the given points are contained in a unit disk, any three disks from C have a nonempty intersection. By Helly's Theorem, all the disks have a nonempty intersection. Let $q$ be a point from the intersection. Then $q$ belongs to every disk from C and is, therefore, at a distance less than $1$ from there centers. In other words, all the centers of the disks from C lie in a disk of radius $1$ centered at $q$.

Now like previous corollary does the follwoing also true ?

Given $s (s > 0)$ points in the plane such that every $6$ or $7$ of them are contained in two disk of radius $1$. Then all $s$ points are contained in two disk of radius $1$.

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Also posted to MO, mathoverflow.net/questions/111442/… --- with no reference on either site to the other site. Gross breach of etiquette. –  Gerry Myerson Nov 4 '12 at 11:43

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