For any set $X$ we set $[X]^2 = \big\{\{x,y\}: x,y \in X, x\neq y\big\}$.


Consider the following statement (S):


**Statement (S)** : Let $G=(V,E)$ be a finite undirected graph and $V\_1, \ldots, V\_n\subseteq V$ with the following properties: 


* $V = V\_1\cup\ldots\cup V\_n$,
* $E \subseteq [V\_1]^2 \cup \ldots \cup [V\_n]^2$, that is every member of $E$ is "inside" some $V\_i$,
* for $i\in\{1,\ldots,n\}$ we have $\chi(G\_i) = n$ where $G\_i = (V\_i, E\cap [V\_i]^2)$, and
* for $i\neq j\in\{1,\ldots,n\}$ we have $[V\_i]^2\cap [V\_j]^2 \cap E = \emptyset$, that is, $V\_i$ and $V\_j$ share no common edge.


**Then** we have $\chi(G) = n$.


(Statement (S) ends here.)


**Questions.**


1. Is (S) provably false?
2. If not, does the [Erdös-Faber-Lovasz conjecture (EFL)](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Faber%E2%80%93Lov%C3%A1sz_conjecture) imply (S)? Note that clearly (S) is a (possibly false) statement implying (EFL).



A stronger version of the Erdös-Faber-Lovasz conjecture?
It is false already for $n=2$. Take a 5-cycle 12345 and sets 123, 3451.


