  In 1981, Kelly showed that planar posets can have arbitrarily large
dimension. However, the posets in Kelly's example have bounded Boolean
dimension and bounded local dimension, leading naturally to the questions as to
whether either Boolean dimension or local dimension is bounded for the class of
planar posets. The question for Boolean dimension was first posed by
Ne\v{s}et\v{r}il and Pudl\'ak in 1989 and remains unanswered today. The concept
of local dimension is quite new, introduced in 2016 by Ueckerdt. Since that
time, researchers have obtained many interesting results concerning Boolean
dimension and local dimension, contrasting these parameters with the classic
Dushnik-Miller concept of dimension, and establishing links between both
parameters and structural graph theory, path-width, and tree-width in
particular. Here we show that local dimension is not bounded on the class of
planar posets. Our proof also shows that the local dimension of a poset is not
bounded in terms of the maximum local dimension of its blocks, and it provides
an alternative proof of the fact that the local dimension of a poset cannot be
bounded in terms of the tree-width of its cover graph, independent of its
height.
