Finite sums of arithmetic progressions

Shahram Mohsenipour

Published 2019-02-26Version 1

For two $l$-term arithmetic progressions $P$ and $Q$, we define their pointwise sum $P\oplus Q$ as the $l$-term arithmetic progression whose $i$th term is the sum of the $i$th terms of $P$ and $Q$. So we can talk about finite sums of $l$-term arithmetic progressions. In this paper we give a combination of van der Waerden's theorem and Hindman's theorem by showing that for positive integers $c$ and $l\geq3$, if $\mathbb{N}$ is $c$-colored, then there are infinitely many $l$-term arithmetic progressions $P_i$, $i\in\mathbb{N}$ such that all of their finite sums (with no repetition) are monochromatic with the same color, call it $\gamma$. We can do more by showing that all the common differences of the finite sums of $P_i$ can have also the color $\gamma$, thus a combination of the van der Waerden-Brauer theorem and Hindman's theorem. We also give a tower bound for the finite case of the first mentioned theorem.