Subposets of small Dushnik-Miller dimension

Here is a slight improvement: by observing that a poset has dimension $\le 2$ if it can be partitioned into two chains, you can improve $\sqrt n$ to $\sqrt {2n}$ for $n>1$. (Let the width be $w$. If $w\ge \sqrt {2n}$ then there is an antichain of size $w\ge \sqrt {2n}$. Otherwise there is a partition into $w$ chains of average size $\frac{n}{w}\ge\sqrt{\frac {n} {2} }$, so that the union of the largest two of these chains has size $\ge\sqrt {2n}$.)

Elyse Yeager and I have constructed examples for an upper bound. Basically, if you understand the dimension of subposets of $P$, then you understand the dimension of the subposets of the lexicographic power $P^k$. Starting with an appropriate standard example then gets you a sublinear upper bound on $F_d(n)$; this bound depends on $d$, and in particular we have $F_2(n)\leq n^{0.8295}$.

Our note is on the arXiv: http://arxiv.org/abs/1404.0021

Here, I think it is better to think about the dimension in terms of the hypergraph of incomparable pairs. For posets of dimension 2, a poset has dimension 2 if and only if the graph of incomparable pairs is bipartite. (see http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.339 for a proof)

Thus you would want to construct posets whose graph of incomparable pairs had no large induced bipartite subgraphs (since a subposet will correspond to an induced subgraph of the graph of incomparable pairs). You could look at the constructions of graphs with no large independent sets (for example, random ones) since those will have no large bipartite subgraphs.

For larger k you can recast the problem as an extremal hypergraph coloring problem and probably there exist some existing hypergraph coloring results?