I am reading a paper where I'm trying to fill in the details in one of the proofs, and I came across the following combinatorial question:
Suppose the following: $M$ is a set with $m$ elements. For some $i \in \lbrace 1, \ldots, m-2 \rbrace$ we have $i+2$ distinct subsets \begin{align*} E_1, E_2, \ldots, E_{i+2} \subset M, \end{align*} each containing $i$ elements. Is it possible, for each $j$, to add an element $p_j \in M \setminus E_j$ to the set $E_j$ to get $F_j := E_j \cup \lbrace p_j \rbrace$ such that the sets $F_1, \ldots, F_{i+2}$ are distinct?
I tried to prove that this is true by counting the number of possible collections $(F_1, \ldots, F_{i+2})$ you can obtain by adding possible $p_j$ to $E_j$ for each $j$, hoping to prove that this number was bigger than the number of collections $(F_1, \ldots, F_{i+2})$ where two of the entries are the same set. However, after some experimenting that doesn't seem to be the case$\ldots$
Any ideas or thoughts are very welcome!