Let $\mathscr{C}$ be a cover of $\mathbf{N}=\{1,2,\dotsc,N\}$ by finite subsets $S\in \mathscr{C}$ with $2\leq |S|\leq K$, where we write $|S|$ for the number of elements of $S$. Assume no element of $\mathbf{N}$ is contained in more than $K'$ elements of $\mathscr{C}$.
Given a subset $Z\subset \mathbf{N}$, we define the boundary $\partial Z$ of $Z$ to be the set of elements $n\in Z$ such that $n+1\not\in Z$. Assume that every $S\in \mathscr{C}$ satisfies $|\partial S|>|S|/2$.
Define a graph $\Gamma$ with elements of $\mathscr{C}$ as vertices, having an edge between $S,S'\in \mathscr{C}$ iff there are $n\in S$, $n'\in S'$ such that $n'\in \{n-1,n,n+1\}$.
Under what conditions is it the case that there must be a subset $\mathscr{D}\subset\mathscr{C}$ such that (a) the subgraph $\Gamma|_\mathscr{D}$ is connected and (b) the boundary of $\bigcup \mathscr{D}$ is large (meaning $\gg N$, say)?
(A satisfactory solution to the case $K'=1$ is given by Existence of connected component with large boundary? In brief, it is then enough for every vertex of $\Gamma$ to have degree $\geq 3$. The condition can be easily relaxed: for $K'=1$, it is enough for $\Gamma$ to have many surviving vertices of degree $\geq 3$ after we repeatedly prune all vertices of degree $1$.)
