Sums, products, and exponents in two-colorings of the naturals

2025/12/10 by Ryan Alweiss, Alweiss, Ryan, Matthew L. Bowen +3
Mathematics · #05D10 #54D80 #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2512.09598

openalex publication_date 2025/12/10 · openalex created_date 2025/12/12 · openalex updated_date 2026/07/28

Abstract

We prove that for any coloring of the naturals using two colors there are monochromatic sets of the form \x,y,xy,x+iy:i≤ k\ and \x,y,x^y,xyⁱ:i≤ k\ for any k.

Citations

• Bounds for monochromatic solutions to \x+y,xy\
• Sums and products in sets of positive density
• Revisiting sums and products in countable and finite fields
• Monochromatic non-commuting products
• Monochromatic Sums and Products of Polynomials
• Monochromatic Solutions to Systems of Exponential Equations
• Combinatorial problems in finite fields and Sidon sets
• On monochromatic solutions of some nonlinear equations in Z/pZ

Related

• Monochromatic products and sums in 2-colorings of ℕ
• Monochromatic Sums and Products with Additive or Multiplicative Shifts in Natural Numbers
• Exponential bounds for monochromatic sums equal to products
• An affirmative answer to Owings's sumset question
• Monochromatic sums and products in ℕ