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Let $n \geq 14$ be a positive integer. In a city there are more than $n$ clubs, all of them have exactly 14 members. At each group of $n+1$ clubs there is a person who is member of at least 15 of these $n+1$ clubs. Show that it is possible to select $n$ people such that all clubs have a member among these people.

This question is from Hungary. I tried to proceed by induction on the number $n$ and I could prove the base case $n=14$. However I don't know how to go on. Any help is welcome.

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  • $\begingroup$ Pigeon-hole principle. $\endgroup$ – Rogelio Molina Jun 5 '15 at 7:12
  • $\begingroup$ @RogelioMolina Could you provide a detailed proof? $\endgroup$ – PSPACEhard Jun 5 '15 at 7:19
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    $\begingroup$ @RogelioMolina How do you use Pigeonhole principle to prove this problem? $\endgroup$ – jack Jun 5 '15 at 16:11
  • $\begingroup$ It is a hunch, I could be wrong, but I will look into the problem later today if I get a chance. I would start by drawing a graph, members as vertices and coloured edges for clubs, one colour for each club. $\endgroup$ – Rogelio Molina Jun 5 '15 at 17:04
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Let us pick any $n$ clubs, and call this the base set of clubs. The rest of the clubs are currently considered uncovered. We also have a (currently empty) solution set $S$, where we aim to have the final set of $n$ members who will cover all of the clubs. Repeat the following process until there are (1) no more uncovered clubs, or (2) we have repeated $n$ times.

  1. Pick any uncovered club.
  2. Apply the given condition on this set of $n+1$ clubs: the base set + the club picked in $(1)$. From the problem statement, we get a member, say $M$, that is in atleast $15$ of these clubs.
  3. Add $M$ to our solution set $S$, and mark as covered, all clubs not in the base set, and having $M$ as a member.

Now there are 2 cases. Either (1) we ran the above steps $k$ times, $k \leq n$ and we have no more uncovered sets. Or (2) we ran the above steps exactly $n$ times, and there are still some uncovered sets.

Case (1): The only problem here is that some of the base sets might still not be covered. Note that each of the $k$ members in $S$ are present in at least $14$ of the base sets. That accounts for at least $14k$ slots out of the available $14n$ slots in the base set. Which means atleast $14k/14 = k$ base sets have been covered, leaving us with at most $n-k$ uncovered base sets. Pick one member from each of these (at most) $n-k$ base sets and add them to our solution set $S$, and we have a valid final solution set.

Case (2): We have exactly $n$ members in our solution. Each of those $n$ members are present in atleast 14 of our base sets, thus taking up atleast $14n$ base slots, which happen to be all of our base slots. Hence if we now consider the following $n+1$ clubs: base set + any uncovered club, we get a contradiction to the given condition in the problem statement.

EDIT: It seems that the choice of $14$ and $15$ in the problem statement is arbitrary, any two consecutive numbers would have served the purpose.

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  • $\begingroup$ Make sense! Great! $\endgroup$ – PSPACEhard Jun 5 '15 at 8:12
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    $\begingroup$ @mathosopher I don't think your solution is true. Let's say for example, $M$ is a member of all $n+1$ clubs (the $n$ clubs from the base set + the club picked in (1)). When you change the club picked in (1) and keep the base set constant, the member $M$ is still a person satisfying the condition of the problem (i.e. $M$ is a member of at least 15 clubs). Repeating this process $n$ times the only member we have is $M$. So your assumption of: '(2) we have repeated $n$ times' => '(2) we have exactly $n$ members in our solution' doesn't seem to be true. $\endgroup$ – jack Jun 5 '15 at 16:09
  • $\begingroup$ @mathosopher Actually your solution can be easily fixed. It's possible to find a group of $n$ clubs such that every person from this group is member of at most 14 from these $n$ clubs (otherwise we are done by induction). We consider this group the base set of clubs. So what I wrote above does not hold anymore. Thanks for your solution! $\endgroup$ – jack Jun 5 '15 at 23:46
  • $\begingroup$ @jack Nice catch! And an even nicer fix! $\endgroup$ – mathosopher Jun 6 '15 at 3:07

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