Eighty-three sublattices and planarity

Abstract

Abstract Let L be a finite n-element lattice. We prove that if L has at least $$83\cdot 2^{n-8}$$83·2n-8 sublattices, then L is planar. For $$n>8$$n>8, this result is sharp since there is a non-planar lattice with exactly $$83\cdot 2^{n-8}-1$$83·2n-8-1 sublattices.