# Finite sums of arithmetic progressions

## TL;DR

This paper provides a combinatorial proof for a generalized theorem related to arithmetic progressions and Hindman's theorem, along with bounds for finite cases.

## Contribution

It introduces a new combinatorial proof for a two-fold generalization of classical theorems and establishes tower bounds for finite versions.

## Key findings

- Purely combinatorial proof of the generalization
- Tower bounds for finite cases
- Extension of van der Waerden-Brauer and Hindman's theorems

## Abstract

We give a purely combinatorial proof for a two-fold generalization of van der Waerden-Brauer's theorem and Hindman's theorem. We also give tower bounds for a finite version of it.