Notes on planar semimodular lattices. IV. The size of a minimal congruence lattice representation with rectangular lattices

Let D be a finite distributive lattice with n join-irreducible elements. In Part III, we proved that D can be represented as the congruence lattice of a special type of planar semimodular lattices of O(n(3)) elements, we called rectangular. In this paper, we show that this result is best possible. Let D be a finite distributive lattice whose order of join-irreducible elements is a balanced bipartite order on n elements. Then any rectangular lattice L whose congruence lattice is isomorphic to D has at least kn3 elements, for some constant k > 0.