KöMaL 2021 A797

We call a system of non-empty sets $H$ entwined, if for every disjoint pair of sets $A$ and $B$ in $H$ there exists $b\in B$ such that $A\cup\{b\}$ is in $H$ or there exists $a\in A$ such that $B\cup\{a\}$ is in $H$. Let $H$ be an entwined system of sets containing the following $n$ one-element sets: $\{1\}, \{2\},\dots,\{n\}$. Prove that if $n>k(k+1)/2$, then $H$ contains a set with at least $k+1$ elements, and this is sharp for every $k$, i.e. if $n=k(k+1)$, it is possible that every set in $H$ have at most $k$ elements.