# A Hales--Jewett type property of finite solvable groups

## TL;DR

This paper proves that finite solvable groups satisfy a Hales--Jewett type property, supporting a conjecture in Euclidean Ramsey theory by linking group symmetry with Ramsey properties of finite sets.

## Contribution

It establishes that all finite solvable groups have a Hales--Jewett type property, advancing understanding of symmetry and Ramsey theory in finite groups.

## Key findings

- Finite solvable groups satisfy the Hales--Jewett type property.
- Supports the conjecture linking group symmetry to Euclidean Ramsey theory.
- Enables recovery of Kříž's work in Euclidean Ramsey theory.

## Abstract

A conjecture of Leader, Russell and Walters in Euclidean Ramsey theory says that a finite set is Ramsey if and only if it is congruent to a subset of a set whose symmetry group acts transitively. As they have shown the ``if" direction of their conjecture follows if all finite groups have a Hales--Jewett type property. In this paper, we show that this property is satisfied in the case of finite solvable groups. Our result can be used to recover the work of K\v{r}\'i\v{z} in Euclidean Ramsey theory.