Dimension versus size
The order dimension of a poset P (dim P) is the least s for which there exist linear extensions \(\sigma_ 1,...,\sigma_ s\) of P (i.e., order preserving bijections \(P\to \{1,...,| P| \})\) such that for all \(p,g\in P\) with \(p\nless g\) there exists \(i\in \{1,...,s\}\) with \(\sigma_ i(p)>\sigma_ i(g)\). The authors investigate also behaviour of f(d), the least size of a lattice of order dimension d. If \(P_ n\) is the lattice of a projective plane of order n then \(\dim P_ n>n/2 \ln (n)\) and \(f(d)=O(d\) 2log 2d). If L is a lattice of height 3 then dim L\(=O(| L|^{1/2} \log | L|)\).
scientific article; zbMATH DE number 4068941