# One hundred twenty-seven subsemilattices and planarity

## TL;DR

This paper establishes a threshold for the number of subsemilattices in finite semilattices that guarantees planarity, providing a sharp bound and characterizing non-planar cases.

## Contribution

The paper proves a precise numerical criterion linking the count of subsemilattices to the planarity of finite semilattices, including sharpness of the bound.

## Key findings

- Semilattices with at least 127·2^{n-8} subsemilattices are planar.
- There exists a non-planar semilattice with 127·2^{n-8}-1 subsemilattices.
- The bound is sharp for all n > 8.

## Abstract

Let $L$ be a finite $n$-element semilattice. We prove that if $L$ has at least $127\cdot 2^{n-8}$ subsemilattices, then $L$ is planar. For $n>8$, this result is sharp since there is a non-planar semilattice with exactly $127\cdot 2^{n-8}-1$ subsemilattices.