# Set of common representatives and pigeonhole principle in one problem

We are given set $\{1, \dots n\}$ and $A = \{A_1 \dots A_s\}$ such as $|A_i|=k$, $|A| = s = \binom n k$, namely $A$ contists of all possible subsets of size $k$.
We say that $S$ is a set of common representatives, if $\forall i \in 1\dots k :S\cap A_i \ne \emptyset$.

Task is to find amount of possible distinct representative sets $S$.
Example $A = \{\{1,4,5\}\{1,2,5\},\{1,2,3\},\{2,3,4\},\{3,4,5\}\}$
Here, $\tau(S) = 5$, since there are two possible set of representatives: $\{1,4\},\{1,3\},\{2,5\},\{3,5\},\{2,4\}$

Solution draft
I figured, that $|S| = n-k+1$ but probably I lack combinatorial experience to apply pigeonhole principle to my problem.
All formulations of it say that that exists at least one box, for which some measure applies, but here I must guarantee that measure applies for all of the boxes.
Yours help would be immensely appreciated.

• You are on the right track. One part, probably the easy one, is to show that any smaller set $S$ is not "a set of common representatives. The other part is to show that if $S$ is at least as big as what you figured, then it is a set of common representatives as you've defined that. Commented Dec 20, 2014 at 18:43
• @hardmath It's easy to show that: if we strip our $S$ for at least 1 element, than, since $s = \binom m k$, we would have one $A_x$ not covered by $S$. And our $S$ is big enough, because $S$ of size $n$ is definitely a set of common representatives, and we may decrease size till it reaches $|n-k+1|$. But you are missing my point. What I am asking is, how many $\underline{\text{different}}$ sets of representatives exists for our $A$. My original guess was $\binom n k \cdot k$ (like we choose one $A_i$ and then choose one element of $A_i$, which will stay in $S$, while others removed) Commented Dec 20, 2014 at 18:53
• If you mean different sets $S$ that satisfy your definition, then the only tweak I'd make to your "solution draft" is to say $|S| \ge n-k+1$. Counting these is a sum of binomial coefficients. Commented Dec 20, 2014 at 18:58
• @hardmath We are only interested in possible representative sets of minimal size. So $\ge$ is not needed. Would you be so kind to comment on my guess for $\tau(S)$ in my previous comment? I'm not sure that it's correct since it don't take into account possible duplicates. Commented Dec 20, 2014 at 19:01
• @hardmath OOh. I guess I get what you are saying. $\tau(S) = \binom n {n-k+1}$ Commented Dec 20, 2014 at 19:08

## 1 Answer

If I understand correctly a set $S$ is of common representatives if every subset of $\{1,2\dots n\}$ that has $k$ elements has an element in common with $S$.

A set $S$ is of common representatives if and only if $S$ has $n-k+1$ elements or more. How many elements have $n-k+1$ subsets? $\sum\limits_{j=n-k+1}^n\binom{n}{j}$.

If you only want those of minimal size the answer is $\binom{n}{n-k+1}$

• Yes, it's correct. We have posted answers almost simultaneously) We are only interested in minimal size $S$, so sum aren't required, but it was mentioned only in comments. Hence, I'm going to accept this. Commented Dec 20, 2014 at 19:10
• Oh I see. thank you very much. Commented Dec 20, 2014 at 19:11