I have very recently started studying graph theory and having difficulty wrapping my mind around the following question from a problem set that I am working on. Most of the similar problems are on Odd town/even town. However, I assume this question, being relevant, is not the same. We have recently learned about maximal-matching, Hall's theorem and some of the other Extremal graph theorems. My guess was that it is asking me to use some of these tools. For example, Should I be approaching this question as a question of maximal/maximum matching? Should I try using Hall's condition/theorem (which I have not understood very well)? Any leads/guidance towards a solution would be greatly appreciated.
A town has $\textit{n}$ residents and $\textit{m}$ clubs $C_1, \ldots C_m$. Each club has $\textit{k}$ members, where $0 \le k < n/2$, and no two clubs have the same set of members.
Prove that it is possible for each club to admit exactly one new member each, in such a way that it is still the case afterwards that no two clubs have the same set of members.
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My very rough draft attempt/brainstorming at it:
Let $n = |V|$ represent the residents and $\textit{m}$ be the number of clubs. Since we are given that all clubs are distinct (no two clubs have the same set of members), we see that $m \le 2^n$ because the set of all possible clubs is power set of $\textit{m}$. We know that for each club $C_i$, the number of members is $\textit{k}$, where $0 \le k < \frac{n}{2}$. We will prove that for each $C_i$ where $|C_i| = k + 1$ such that no two clubs have the same set of members.
If $\textit{n = 1}$, then we know that if we have more than 1 club, no clubs have any member because we know that each club has to have $\textit{k}$ members and $k < n/2$. If $\textit{m = 1}$, then we are good and we can add exactly one more member to the club. ....
Let the residents be the set $R=\{1, \ldots, n\}$. Let $X$ be the set of all the subsets of $R$ of size $k$ and $Y$ be the set of all the subsets of $R$ of size $k+1$.
Define a bipartite graph $G=(X, Y)$ by joining an edge between $A\in X$ and $B\in Y$ if $A\subseteq B$. The degree of each vertex in $X$ is $n-k$ and the degree of each vertex in $Y$ is $k+1$. We show that there is a matching in $G$ that saturates $X$ by verifying the Hall's matching condition. Let $S\subseteq X$. Then by double counting, we have $|S|(n-k)\leq |N(S)|(k+1)$. Using $k<n/2$ we see that $|N(S)|\geq |S|$ and the Hall's condition is satified. So there is a matching saturating $X$.
Can you finish?