Let $S \subset 2^M$ be a family of subsets of a set $M$ with $|M| = n$. Call $S$ min-k-intersecting if for each pair of subsets the intersection has at most $k$ elements: $\forall A,B \in S: |A \cap B | \leq k$.

Question: What is the maximal $N(n,k) = |S|$ of such a min-k-intersecting family of subsets of a set with cardinality $n$?

I'd be glad if someone could point me to an argument (e.g. of Erdos-Rado type) that would be applicable.

Thanks a lot.

Let $S \subset 2^M$ be a family of subsets of a set $M$ with $|M| = n$. Call $S$ min-k-intersecting if for each pair of subsets the intersection has at most $k$ elements: $\forall A,B \in S: |A \cap B | \leq k$.

Question 1: What is the maximal $N(n,k) = |S|$ of such a min-k-intersecting family of subsets of a set with cardinality $n$?

Question 2: What is the maximal $N(n,k,m) = |S|$ if additionally $|A| = m \quad \forall A \in S$?

I'd be glad if someone could point me to an argument (e.g. of Erdos-Rado type) that would be applicable.

Thanks a lot.