# The existence of monochrome simplexes of a given volume on a rational lattice colored in $n$ colours

## TL;DR

This paper proves that in a finite-colored multidimensional rational lattice, there always exists a monochrome simplex of any specified volume, extending geometric and combinatorial understanding of lattice colorings.

## Contribution

It establishes the existence of monochrome simplexes of any volume on rational lattices with finite colorings, a novel result in geometric combinatorics.

## Key findings

- Monochrome simplexes of any volume exist on rational lattices with finite coloring.
- The result applies to multidimensional rational lattices.
- It advances understanding of lattice colorings and geometric configurations.

## Abstract

The paper proves the existence of monochrome standard simplexes of a given volume on a multidimensional rational lattice painted in a finite number of colors.