Posets and planar graphs

Usually dimension should be an integer valued parameter. We introduce a refined version of dimension for graphs, which can assume a value 1⁄2 t 1l t , thought to be between t 1 and t. We have the following two results: (a) a graph is outerplanar if and only if its dimension is at most 1⁄22l3 . This characterization of outerplanar graphs is closely related to the celebrated result of W. Schnyder [16] who proved that a graph is planar if and only if its dimension is at most 3. (b) The largest n for which the dimension of the complete graph Kn is at most 1⁄2t 1l t is the number of antichains in the lattice of all subsets of a set of size t 2. Accordingly, the refined dimension problem for complete graphs is equivalent to the classical combinatorial problem known as Dedekind’s problem. This result extends work of Hoşten and Morris [14]. The main results are enriched by