# Eighty-three sublattices and planarity

## TL;DR

This paper establishes a threshold for the number of sublattices in a finite lattice that guarantees its planarity, providing a sharp bound with a specific constant.

## Contribution

The authors prove a new quantitative criterion linking the number of sublattices to planarity in finite lattices, with a sharp bound for all sizes.

## Key findings

- Lattices with at least 83·2^{n-8} sublattices are planar.
- The bound is sharp for n > 8, with a non-planar lattice having just below this number.
- The result precisely characterizes the sublattice count threshold for planarity.

## Abstract

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