# Covering inequality for sets of intervals

Let $I$ and $J$ be finite sets of open intervals $(a,b)\subset\mathbb R$. For a finite set of points $P\subset \mathbb R$ we denote those subsets of intervals from $I$ and $J$ containing some point from $P$ by $I_P,J_P$. Now suppose that \begin{align}\tag{*}\label{IP JP ineq}\lvert I_P\rvert\le \lvert J_P\rvert+1\end{align} for all finite subsets $P$, and that the inequality \eqref{IP JP ineq} is optimal in the sense that there exists at least one finite $P$ for which equality is achieved. Here $\lvert\cdot\rvert$ is simply the counting measure.

I have the strong suspicion (supported by numerical experimentation with random sets) that there has to exist some specific interval $(a,b)\in I$ such that $(a,b)$ contains at least one point from each set $P$ for which equality is achieved in \eqref{IP JP ineq}.

It is clear that the statement cannot be true for more general subsets than intervals but I couldn't come up with any argument yet. It is also clear that the claim cannot hold true if the $1$ in \eqref{IP JP ineq} is replaced by $0$.