Suppose $n$ is a big number and $k\geq 2$. How many sets $S\_1,\dots,S\_m\subset [n]$ can we find such that
(1) $|S\_i| = k$ for all $i$,
(2) $|S\_i\cap S\_j| \leq 1$ for all $i\ne j$.
What's the maximum possible value of $m$? 


(I just need to know the growth order of $m$ depending on $n$ and $k$. For instance, when $k=2$, we have $m = \binom{n}{2} \sim n^2$.)


I tried to look it up in the literature, but it looks like this is different from the classical intersecting family that I have an upper bound on the size of intersection of a pair of sets instead of a lower bound.



maximum size of intersecting set families
I noted the simple upper bound $m\le \binom{n}{2}/\binom{k}{2}= n(n-1)/(k(k-1))$ in my comment above. It seems that Wilson proved that if $k-1$ divides $n-1$, and $\binom{k}{2}$ divides $\binom{n}{2}$ then for large $n$ this upper bound is attained. See page 1424 of this interesting ICM article of Péter Frankl's ICM article [*Intersection Theorems for Finite Sets and Geometric Applications*](https://pdfs.semanticscholar.org/f3d8/5b21424bae8e73f9ad94277059f83b98cad4.pdf), which discusses many such related problems.


More generally, let $m(n,k,t)$ denote the size of the largest collection of $k$ element subsets of $\{1,\ldots, n\}$ such that any two sets intersect in at most $t-1$ elements. (Or equivalently, every $t$ element set is a subset of at most one set from our collection.) The simple argument in my comment gives that $m(n,k,t)\le \binom{n}{t}/\binom{k}{t}$. Proving a conjecture of Erdős and Hanani, Rödl proved that for fixed $k$ and $t$ and as $n\to \infty$ one has 
$$ 
m(n,k,t) \sim \frac{\binom{n}{t}}{\binom{k}{t}}.
$$
Rödl's paper (which is known as the nibble method) is [here](http://ac.els-cdn.com/S0195669885800238/1-s2.0-S0195669885800238-main.pdf?_tid=f54a0e4c-b2d2-11e3-8c9e-00000aab0f26&acdnat=1395610668_3702554a6f59cc9472eb480547147639).


