The starting point of this question is the fact that for some simple, undirected graphs $G, H$ there is no graph homomorphism $f:G\to H$. This is the case for instance if $\chi(G)>\chi(H)$.

Informally speaking, the bit below is about how "close" a map between the vertex sets of graphs can become to being a graph homomorphism.

More formally, given (finite or infinite) graphs $G, H$ and a **function** $f:V(G)\to V(H)$ we say that $e\in E(G)$ is *faulty* with respect to $f$ if $\text{im}(f|_e) = f(e) \notin E(H)$, be it because $f(e)$ only consists of $1$ element, or be it because $f(e)$ consists of two non-adjacent vertices of $H$.

Let $\text{Flt}(G, H)$ consist of those sets $T\subseteq E(G)$ such that there is a map $f:G\to H$ such that $T$ is the collection of faulty edges with respect to $f$. We call the members of $\text{Flt}(G,H)$ *faulty edge sets*.

**Question.** When we consider the poset $(\text{Flt}(G,H),\subseteq)$, does every faulty edge set contain a minimal faulty edge set?