The Dushnik–Miller dimension of a partial order $(P,{\leq})$ is the smallest possible size $d$ for a family ${\leq_1},\ldots,{\leq_d}$ of total orderings of $P$ whose intersection is ${\leq}$, i.e. $x \leq y$ iff $x \leq_i y$ holds simultaneously for all $i = 1,\ldots,d$. Equivalently, the dimension is the smallest $d$ such that $P$ embeds in $L^d$ of some total ordering $L$, where $L^d$ is endowed with the coordinatewise partial ordering.

Since chains have dimension 1 and antichains have dimension 2, Dilworth's Theorem guarantees that every poset of size $n$ contains a $2$-dimensional subposet of size at least $\sqrt{n}$. Is this optimal? In general, what can we say about subposets of dimension $d$?

Tom Goodwillie's argument below shows that for sufficiently large $n$, every poset of size $n$ either has an antichain of size $\sqrt{dn}$ or a $d$-dimensional subposet of size $\sqrt{dn}$. This result is optimal for $d = 1$; stated this way, this could also be optimal for $d > 1$ too. For $d = 2$, this improves my lower bound $\sqrt{n}$ above by a factor of $\sqrt{2}$.

In view of this, let me reformulate the question as follows. Let $F_d(n)$ be the largest integer such that every poset of size $n$ has a $d$-dimensional subposet of size $F_d(n)$. Note that $F_1(n) = 1$ for all $n$ and, when $d > 1$, $F_d(n) \geq \sqrt{dn}$ for large enough $n$.

Is $F_2(n) \leq C\sqrt{n}$ for some constant $C$? In general, what is the asymptotic behavior of $F_d(n)$?