Points inside unitary circle

Question

Let $A$ be a set of $25$ points from the plane which has the property that given any three elements of $A$, two of these are less than $1$ apart. Show that there is a circle of radius $1$ that contains at least $13$ points from $A$.

Answer 1

If all points are less than 1 unit apart, then the conclusion holds trivially.

If not, then pick 2 points a and b such that they're not less than a unit apart. Then for all the other 23 points each of them must either be less than 1 unit away from a or b by our assumption. This means either a or b must have at least 12 points that are less than 1 unit away from them. (22/2 = 11) Without loss of generality assume this is the case for a, then draw a unit circle centered at a which by definition will contain at least 12 points plus a itself making it at least 13 points.