Let $A\_1, A\_2, \ldots, A\_n$ be $n$ sets such that:


(1) for each $i\in [n]$, $\frac{n}{3}\leq |A\_i|\leq n$;


(2) for any $1\leq i<j<k\leq n$, $|A\_i\cap A\_j\cap A\_k|\leq a$, where $a$ is a constant and $n$ is sufficiently large.


What is $\min |A\_1\cup A\_2\cup \cdots \cup A\_n|$? Is it $\Omega(n^2)$ or $o(n^2)$?


Remark 1. If we change condition (2) to $|A\_i\cap A\_j|\leq a$ for every $i\neq j$, then the problem is related to Corradi's lemma (in the language of hypergraph) <https://de.wikipedia.org/wiki/Lemma_von_Corr%C3%A1di>


Remark 2. I can prove $\min |A\_1\cup A\_2\cup \cdots \cup A\_n|=\Omega(n^{\frac{3}{2}})$ for any constant $a$ using some double counting arguments. (Hint: Let $A=A\_1\cup A\_2\cup \cdots \cup A\_n$, and for any $x\in A$, let $d(x)=|\{A\_i\colon\, x\in A\_i, i\in [n]\}|$. Then $\sum\_{x\in A}d^3(x)=\sum\_{(i,j,k)\in [n]^3}|A\_i\cap A\_j\cap A\_k|$.)



n sets, each is large, the intersection of every three is small, what is the size of the union?
Let $m$ be chosen later, and let $A\_1, A\_2, \dots, A\_n$ be independently chosen random subsets of $\{1,2,\dots m\}$, each having size $n$.


For a fixed $a+1$-tuple $(x\_1, x\_2, \dots, x\_{a+1})$ of distinct elements from $\{1,\dots,m\}$, and a fixed triple $(i,j,k)$, the probability that $\{x\_1, \dots, x\_a\} \subseteq A\_i \cap A\_j \cap A\_k$ is at most $\left(\frac{n}{m}\right)^{3a+3}$. Taking the union bound over all $x\_1, x\_2, \dots, x\_{a+1}$ and all $(i,j,k)$, the probability that there is some collection of $a+1$ elements in the intersection of some $3$ sets is at most
$$m^{a+1} n^3 \left(\frac{n}{m}\right)^{3a+3} = n^{6+3a} m^{-2a-2}$$
In particular, if $m=n^{\alpha}$ and $\alpha>\frac{6+3a}{2a+2} = \frac{3}{2}\left(1+\frac{1}{a+1}\right)$, there is a positive probability that none of the intersections is larger than $a$, so a collection of subsets of $[m]$ must exist with no large intersections.


This gives an $o(n^2)$ bound for $a \geq 3$.


