🚀 go-pugleaf

Is it true that every partially ordered set of height 2 is representable
by subspaces of a 6-dimensional vector space?

Here:

   Height 2 means that every chain has at most three elements,
   A partially ordered set T is representable in a vector space V if there
exists a 1:1 mapping from T to the set of subspaces of V which preserves the
order.

In particular, every finite geometry is a partially ordered set of height
2. Is it possible to represent every finite geometry in a 6-dimensional
space (points should go to lines, lines should go to 3- or 2-dim. subspaces).

There exists a partially ordered set (even a semilattice) of height 2 which 
can not be represented in a 5-dim. space. I could not find this example in
the literature so I had to construct it myself.

I know that there is a lot of results in algebraic topology, in the theory of
representations of associative algebras, in combinatorics etc concerning 
representations of partially ordered sets. It would be strange if nobody
knows the answer to my question.

Mark V. Sapir