I have a discrete math problem I've stumbled onto during my research, but I'm not sure I have the math background to solve it. The problem is as follows:
There is a set, $C$, of $n$ elements, where $n = 3t+1$ for some integer $t$.
There are $n$ subsets of $C$, called $C_1 ... C_n$. Each subset contains $n-t$ elements of $C$, and the exact same $n-t$ elements never appear in more than $t$ subsets.
Each subset $C_t$ modifies itself by setting its content to the union of $n-t$ of the subsets of $C$ - this union must include $C_t$. Call the modified subsets $C_1'...C_n'$.
Prove that the intersection of all the modified subsets, $C_1' \cap C_2' \cap ... C_n'$, always has a cardinality of at least $n-t$. (or find a counterexample)
Can someone provide help or at least a starting point?