Denote $[n]=\{1,2,\dots,n\}$. Assume $n\geq2$.



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> **Question.** Is it true that given any $S\_1,S\_2,\dots,S\_{2n}$ (repetition allowed) subsets of $[2n]$ with $a\in S\_a$ and $\# S\_a=n$ for all $1\leq a\leq 2n$, there exist $i, j, k\in[2n]$ (not all equal) such that 
>  $$i\in S\_j, \qquad j\in S\_k, \qquad k\in S\_i \,\, ?$$
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**EDIT.** There was an unfortunate typo: $\#S\_a=n$ instead of $\#S\_a=a$. Sorry.



Existence of 3-distributed subsets
For $n>2$ this is true. Consider a directed graphs with arrows from $a$ to $S\_a\setminus a$. If it has arrows $a\to b$, $b\to a$, take $i=j=a$, $k=b$. If not, consider a vertex $a$ with maximal indegree, it is at least $n-1$ (since all outdegrees are equal to $n-1$). Let $B=\{i\ne a:a\in S\_i\}$, $C=S\_a\setminus a$. Now $B,C$ are disjoint and each of them contains at least $n-1$ vertices. If there is an arrow $c\to b$ $C$ to $B$, we may take $i=c,j=a,k=b$. If not, all arrows from $C$ go to $D=V\setminus (B\cup a)$, but this set contains at most $n$ elements and this may happen only if $S\_c=D$ for all $c\in C$. It gives two opposite edges for $n>2$.


