By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\}: a\neq b \in V\}$. A *graph homomorphism* between graphs $G, H$ is a map $f:V(G)\to V(H)$ such that $\{v, w\}\in E(G)$ implies $\{f(v), f(w)\} \in E(H)$.

If $G,H$ are graphs and there is a graph homomorphism $f:G\to H$ we write $G\to H$, and otherwise $G\not\to H$.

Let $C$ be the set of graphs such that $V(G)=\mathbb{N}$. We set $$E = \big\{\{G,H\}: (G,H\in C) \land (G\not\to H) \land (H\not\to G)\big\}.$$ Let $G_{\mathbb{N}} = (C,E)$.

**Question**: Does $G_{\mathbb{N}}$ have an uncountable clique?