Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T14:13:48.251Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Number of Monochromatic Solutions of ${\bm x}+{\bm y}={\bm z}^{{\bm 2}}$

Published online by Cambridge University Press:  03 January 2006

AYMAN KHALFALAH  and
ENDRE SZEMERÉDI
AYMAN KHALFALAH
Affiliation:
Faculty of Engineering, PO Box Alexandria 21544, Egypt (e-mail: [email protected])
ENDRE SZEMERÉDI
Affiliation:
Department of Computer Science, Rutgers, State University of NJ, New Brunswick, NJ 08903, USA (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In the present work we prove the following conjecture of Erdős, Roth, Sárközy and T. Sós: Let $f$ be a polynomial of integer coefficients such that $2|f(z)$ for some integer $z$. Then, for any $k$-colouring of the integers, the equation $x+y=f(z)$ has a solution in which $x$ and $y$ have the same colour. A well-known special case of this conjecture referred to the case $f(z)=z^2$.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 15 , Issue 1-2 , January 2006 , pp. 213 - 227
Copyright
2006 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)