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

• Published in: Combinatorics, probability & computing Vol. 15; no. 1-2; pp. 213 - 227
• Main Authors: KHALFALAH, AYMAN, SZEMERÉDI, ENDRE
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.01.2006
• ISSN: 0963-5483; 1469-2163
• DOI: 10.1017/S0963548305007169

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 $f(z)=z^2$.