Let $\kappa$ be a cardinal and let $\textrm{Grph}(\kappa)$ be the set of graphs $G = (V,E)$ such that $V \subseteq \kappa$ and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\} \subseteq V: a\neq b\}$.
We define a pre-ordering relation $\to$ on $\text{Grph}(\kappa)$ by \begin{eqnarray}G\to H :\Leftrightarrow \textrm{ there is a graph homomorphism } f:G\to H.\end{eqnarray}
We write $G \leftrightarrow H$ if $G\to H$ and $H\to G$. Note that $\leftrightarrow$ is an equivalence relation and call the equivalence classes the homomorphism equivalence (hom-eq) classes. For finite $\kappa$ the following statements can be proved (see Propositions 2.1. and 2.2. of this):
- Every hom-eq class contains a unique (up to isomorphism) element of smallest cardinality, called the core of that hom-eq class;
- If $G$ is a core of a hom-eq class, then every graph homomorphism $f:G\to G$ is an isomorphism.
Do these statements stay valid in $\textrm{Grph}(\kappa)$ for infinite $\kappa$?
