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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 $A$.

Cgomes
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    Show your own work, lest your question will continue to be downvoted and, perhaps, even closed. Your tagging is a good hint. – DonAntonio Feb 18 '22 at 14:45

1 Answers1

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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.

Dani007
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  • This question seems not to meet the standards for the site. Instead of answering it, it would be better to look for a good duplicate target, or help the user by posting comments suggesting improvements. Please also read the meta announcement regarding quality standards. – Snaw Feb 24 '22 at 02:07