# Rainbow triangles

## TL;DR

This paper investigates conditions in colored vector spaces, especially over finite fields, that ensure the existence of a unit equilateral triangle with vertices in different color classes.

## Contribution

It introduces new conditions for guaranteed rainbow triangles in vector spaces over finite fields, extending classical geometric coloring problems.

## Key findings

- Identifies specific conditions for rainbow triangles in finite field vector spaces.
- Establishes bounds on colorings that guarantee the existence of such triangles.
- Provides new insights into geometric configurations in colored vector spaces.

## Abstract

We seek conditions under which colorings of various vector spaces are guaranteed to have a copy of a unit equilateral triangle, having each vertex in a different color class. In particular, we explore the analogous question in the setting of vector spaces finite fields, with an appropriate notion of distance.