Embedding between Two posets

Question

Suppose $X$ is a finite poset (partially ordered set). Then does an embedding $f$ always exist between the finite poset $X$ and the $\Bbb R^3$ (ordinary $3$-d space)?

Answer 1

Let $\langle P,\le\rangle$ be a partial order in $\Bbb R^3$. For $k=1,2,3$ let $\le_k$ be the partial order on $P$ induced by the natural order on the $k$-th coordinate. Clearly $\langle x_1,x_2,x_3\rangle\le\langle y_1,y_2,y_3\rangle$ iff $x_k\le y_k$ for $k=1,2,3$, so $\le$ is the intersection of the orders $\le_k$ ($k=1,2,3$). Thus, the dimension of $P$ is at most $3$. It follows that the $8$-element partial order of dimension $4$ illustrated in the Wikipedia article on order dimension cannot be embedded in $\Bbb R^3$.