Infinite monochromatic patterns in the integers

TL;DR

The paper proves the existence of infinite monochromatic patterns in integers formed by symmetric polynomials, using ultrafilter algebra, extending classical combinatorial results.

Contribution

It introduces new infinite monochromatic patterns in integers based on symmetric polynomials and employs ultrafilter algebra for the proofs.

Findings

Existence of infinite monochromatic polynomial patterns in integers.

Patterns include sequences and polynomial combinations like sums and products.

Proofs utilize algebraic methods in ultrafilter spaces.

Abstract

We show the existence of several infinite monochromatic patterns in the integers obtained as values of suitable symmetric polynomials. The simplest example is the following. For every finite coloring of the natural numbers $\mathbb{N}=C_1\cup\ldots\cup C_r$, there exists an increasing sequence $a<b<c<\ldots$ such that all elements below are monochromatic, that is, they belong to the same $C_i$: $$a,b,c,\ldots, a+b+ab, a+c+ac, b+c+bc,\ldots,a+b+c+ab+ac+bc+abc,\ldots.$$ The proofs use algebra in the space of ultrafilters $\beta\mathbb{Z}$.