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 $\sum_{x\in A}d^3(x)=\sum_{(i,j,k)\in [n]^3}|A_i\cap A_j\cap A_k|$.)

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