I got this question from my professor and I have no idea how to solve this, any help would be appreciated
"12 politicians give speech every day. Everyone of them uses the same set of 12 arguments. They use different one every day and make sure they won't use the same argument as somebody else already used that day. Using Hall's theorem prove that everyone of them will be able to take part in the 9th day of election without using the same argument twice or using an argument that their opposition used that day"
Call the 12 politicians set $P$ and the 12 arguments set $A$. A one-to-one matching is a function that assigns every politician to at least one argument, and no argument to more than one politician.
Hall's Marriage Theorem is that a one-to-one matching exists iff for every $S$ subset of $P$, the number of elements in $A$ to which $S$ is linked is at least the cardinality of $S$.
In period $t=0$, there's a one-to-one matching, since every politician knows 12 unused arguments, so every subset of the politicians satisfies Hall's. Once arguments are assigned, every politician knows 11 unused arguments, and every subset of politicians still satisfies the hypotheses of the marriage theorem.
In period $t\le 11$, suppose that no politician has repeated an argument nor used the same argument twice in the same day. That means that at period $t \le 11$, each subset of politicians knows $12-t$ unused arguments and each argument has been used in total by $t$ politicians (or otherwise, an argument was made by multiple politicians on the same day). That implies for any subset of politicians $P' \subseteq P$, $|P'| \le |P'|(12-t)$, so there are enough arguments to cover $P'$. So Hall's theorem is satisfied up to period $t=11$, and in period $t=12$ it fails.