In 1977, Trotter and Moore proved that a poset has dimension at most 3
whenever its cover graph is a forest, or equivalently, has treewidth at most
1. On the other hand, a well-known construction of Kelly shows that there are
posets of arbitrarily large dimension whose cover graphs have treewidth 3. In
this paper we focus on the boundary case of treewidth 2. It was recently
shown that the dimension is bounded if the cover graph is outerplanar (Felsner,
Trotter, and Wiechert) or if it has pathwidth 2 (Bir\'o, Keller, and Young).
This can be interpreted as evidence that the dimension should be bounded more
generally when the cover graph has treewidth 2. We show that it is indeed the
case: Every such poset has dimension at most 1276.Comment: v4: minor changes made following helpful comments by the referee