# Collections of sets without $r$-unions covering another set

Is the following known? It seems related to codes and/or Ramsey theory.

Given $$r$$, for what values of $$n,k$$, does there exist a collection of $$n$$ sets whose union contains $$k$$ elements such that none of these sets is contained in the union of at most $$r$$ of the sets.

For example given $$r<3$$ choosing $$n=3$$, $$k=4$$ satisfies the requirement, because of $$\{\{1,2\},\{2,3\},\{3,4\}\}$$. More generally if $$r=2$$ then any $$n$$ and $$k>n$$ satisfies the requirement by generalizing this example.

For $$r=1$$ with any $$n$$, and $$k>\lceil \log_2(n)\rceil$$ one can satisfy the requirement by numbering the sets in order and using the binary representation of their number (like a Hamming code). This argument can be used to show (information theoretic) lower bounds of $$k>\lceil r \log_2(n)\rceil$$.

Let $$f_r(n,k)$$ be the maximum number of $$k-$$subsets of an $$n$$-set satisfying the condition above. Then $$f_r(n,r(t-1)+1+d)\leq \frac{\binom{n-d}t}{\binom{k-d}t},$$ for $$n$$ sufficiently large whenever $$d=0,1$$ or $$d\leq \frac{r}{2t^2}$$ with equality holding iff there exists a Steiner Triple System $$S(t,r(t-1)+1,n-d)$$.