Given any poset $(P,\leq)$ we define the "direct neighbor graph" as follows. Let $$E_P = \big\{\{a,b\}: (a<b \text{ or } a>b) \text{ and } \; ]\min\{a,b\},\max\{a,b\}[ = \emptyset\big\}.$$ It is easy to see the $(P,E_P)$ is a simple undirected graph that can contain a $3$-clique, but not a $4$-clique. But can $\chi(P,E_P)$ become arbitrarily large?

**EDIT.** Peter Taylor points out that my remark on $3$-cliques is wrong - sorry about that.