You want to print a deck of cards of the following type: Each card shows $k$ items out of $n$ different items such that any two cards in the deck share exactly one item.
Question: What is the biggest size of such a deck ?
If you replace "exactly one" by "at least one", this is the classical Erdős-Ko-Rado problem, so I expect this question to be discussed before.
I appreciate both an explanation and any references discussing it.
Also, I have been computing a (very) few numbers and they appear to not be listed in the OEIS, though it is an array in $n$ and $k$, so there are variances how to search the OEIS for it.
Background: such decks of cards exist as a children's two-player game where both players open one card each and the player first naming the common item gets both cards. The player getting all cards wins.