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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: akhalfal@cs.rutgers.edu)
ENDRE SZEMERÉDI
Affiliation:
Department of Computer Science, Rutgers, State University of NJ, New Brunswick, NJ 08903, USA (e-mail: szemered@cs.rutgers.edu)
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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

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