maximum size of intersecting set families 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.
 A: 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, 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.
