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Monochromatic sums and quotients in $\mathbb N$

Mauro Di Nasso, Lorenzo Luperi Baglini, Rosario Mennuni, Mariaclara Ragosta, Alessandro Vegnuti

Abstract

We prove partition regularity of the configuration $x,y,x+y,y/x$ in a strong infinitary form that extends Hindman's Theorem. We study the related issue of partition regularity of configurations involving products of a degree one polynomial in $x$ with one in $y$, reducing the general problem to a handful of special cases.

Monochromatic sums and quotients in $\mathbb N$

Abstract

We prove partition regularity of the configuration in a strong infinitary form that extends Hindman's Theorem. We study the related issue of partition regularity of configurations involving products of a degree one polynomial in with one in , reducing the general problem to a handful of special cases.
Paper Structure (10 sections, 24 theorems, 56 equations)

This paper contains 10 sections, 24 theorems, 56 equations.

Table of Contents

  1. Sums, products and colours.
  2. Results.
  3. Methodology.
  4. Funding
  5. Quotients of sums
  6. A quick review of nonstandard methods
  7. Products of two linear polynomials
  8. Several products of two linear polynomials
  9. Open problems
  10. Appendix: some standard proofs

Key Result

Theorem 1

For every finite colouring $\mathbb N={C}_{1}\cup\ldots\cup{C}_{r}$ there are $m\le r$ and an increasing sequence $(x_i)_{i\in \mathbb N}$ such that, for all $k<\ell\in \mathbb N$ and all ${i}_{1}<\ldots<{i}_{\ell}$, the colour $C_m$ contains all

Theorems & Definitions (69)

  • Theorem 1: \ref{['thm:fsandratios']}
  • Theorem 2: \ref{['thm:squares']}
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Remark 1.2
  • proof
  • proof
  • Theorem 1.4
  • ...and 59 more