TITLE: Large Intersecting Subsets of a Set
QUESTION [10 upvotes]: I am sure the answer to this question is well known, but I am not able to figure it out.
Question: Let $U$ be a finite set. Let $F=(S_1,S_2,...,S_n)$ be such that:
(1) $S_i\subset U$
(2) $|S_i|=n$  
(3) $|S_i\cap S_j|\leq n/2$
Then, what is the lowerbound on $|U$|? In other words, what is the smallest $U$ for which there exists an $F$ satisfying the above conditions.
Clearly, if $U$ has size $n^2$, it easy to construct such an $F$. You can also do this with just $n^2/2$ elements in $U$. Can you do this with just $O(n)$ elements? What about $O(n^{1+\epsilon})$ for a constant $\epsilon<1$?

REPLY [3 votes]: Just to have something for all sufficiently large n, one can also take, for any small enough $\varepsilon>0$, the set $U$ to have $(2+5\varepsilon)n$ points, and then let $S'_i$ be independent random subsets of $U$ each obtained by choosing the points independently with probability $1/2-\varepsilon$. An application of Chernoff's inequality says that the size of any given $S'_i$ is concentrated close to expectation, and in particular greater than $n$. And the expected intersection of any two sets is $(1/4-\varepsilon+\varepsilon^2)(2+5\varepsilon)n$ which is smaller than $n/2$, and also by Chernoff we have good concentration. The failure probability of each of these applications of Chernoff is something like $2^{-\varepsilon^2 n}$. In particular we can certainly take a union bound over all $n+n^2$ applications (in fact we could have exponentially many $S_i$). Alternatively we can take $\varepsilon$ to be something like $\sqrt{\log n/n}$ and get $n$ sets.
So let $S_i$ be a subset of $S'_i$ of size $n$ for each $i$ and the construction is done.