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.



Posets as graphs with the direct neighbor relation
Yes, the chromatic number of a [Hasse diagram](https://en.wikipedia.org/wiki/Hasse_diagram) can be arbitrarily large. Bollobás shows in the following article that for any $k$ there exists a finite lattice whose Hasse diagram is not $k$-colorable.



```
MR505730 05C15 (06A20) 
Bollobás, Béla Colouring lattices. Algebra Universalis 7 (1977), no. 3, 313–314.

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Here are a few other papers that solve the problem, but I have not been able to find any freely available paper.



```
MR748915 05A17 (06B99 06D99) 
Nešetřil, Jaroslav; Rödl, Vojtěch Combinatorial partitions of finite posets and lattices—Ramsey lattices. Algebra Universalis 19 (1984), no. 1, 106–119.

MR1129613 05C15 (06A07) 
Kříž, Igor; Nešetřil, Jaroslav Chromatic number of Hasse diagrams, eyebrows and dimension. Order 8 (1991), no. 1, 41–48.

```

