Fix $c\in (0,1]$ and $r\in\mathbb{N}$. For $n\geq 2r$, define $$f_{r,c}(n):=\max{|\Omega_{n,c,r}|}$$ where $\Omega_{n,c,r}$ is a family of subsets of size $r$ of $(1,2,\ldots ,n)$, such that for every $A\in\Omega_{n,c,r}$, at least $c|\Omega_{n,c,r}|$ elements of $\Omega_{n,c,r}$ intersect $A$.
The Erdos-Ko-Rado theorem settles the case $c=1$: $$f_{r,1}(n)=\binom{n-1}{r-1}$$
For a generic $c$, can we find an explicit formula or an asymptotic for $f_{r,c}(n)$?
