Number of $k$-sets in $[n]$ so that any two of them share at most 2 elements?

Question

Let $[n]$ be $\{1,2,\dots,n\}$. Let \begin{align}T_\ell:=\Big|\Big\{&\{K_1,\dots,K_m\} \text{ is "maximal"}:\\ &\text{each $K_i$ is a $k$-subset of $[n]$, and $|K_i\cap K_j|\le \ell$ for all $1\le i<j\le m$} \Big\}\Big|,\end{align} where "maximal" means you cannot find one more $k$-subset $K_{m+1}$ so that $\{K_1,\dots,K_{m+1}\} \text{ satisfies that each $K_i$ is a $k$-subset of $[n]$, and $|K_i\cap K_j|\le \ell$ for all $1\le i<j\le m+1$}$.

We can assume $1\ll k\ll n$. I am interested in an upper bound on $T_2$. It seems this problem is related to partition, especially when $\ell=0$ ?