Consider $[n] = \{1,\ldots,n\}$. Let $\mathcal{F}$ be a family of subsets of $[n]$ such that every set in $\mathcal{F}$ has exactly $k$ elements and for any $A,B \in \mathcal{F}$ with $A \neq B$, $|A \cap B| \leq r$. Denote the largest possible value of $|\mathcal{F}|$ by $m(n,k,r)$. I'm interested in the special case when $k=2k' \leq \frac{n}{2}$ and $r = k'-1$.
My main question is as follows. What are some good lower bounds for $m(n,2k',k'-1)$ with $k' \geq 3$? Can it be exactly determined in any cases?
Some related questions and results are mentioned below.
In the case $k'=2$, this boils down to evaluating $m(n,4,1)$ which is equivalent to the $K_4$-packing number of $K_n$. This was determined exactly in the paper 'Optimal packing of $K_4$'s into a $K_n$' by A.E Brouwer in 1979.
As noted in this question which links to this paper, the general case is equivalent to finding a lower bound on the maximum number of binary codewords of length $n$ and weight $k$ such that the Hamming distance between any two codewords is at least $2k-r$. Adapting the notation in the paper, I want a lower bound on $A(n,2k-r,k)$. Theorems 4 and 7 do provide some lower bounds but it seems to me that the argument only works if $2k-r$ is even. Moreover, I am interested in the case $A(n,2k-r,k) = A(n,3k'+1,2k')$ and was wondering if you can somehow obtain stronger bounds in this case. (Note that in my case, the results from the paper seem to apply only if $3k'+1$ is even, or equivalently if $k'$ is odd).
This question is also related and provides a nice lower bound attributed to Frenkl, but I could not find a reference to the original statement of this theorem despite some digging around.
The paper 'On a Packing and Covering Problem' by Rödl provides the asymptotic behavior of $m(n,k,r+1)$ (Warning: there is some difference in notation regarding $r$ and $r+1$). Statements such as 'the lower bound holds for sufficiently large $n$' are not helpful to me unless I know what this large $n$ is.
Apologies for the lengthy question! Thank you.