### Abstract

In the present work we prove the following conjecture of Erdos, 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}.

Original language | English |
---|---|

Pages (from-to) | 213-227 |

Number of pages | 15 |

Journal | Combinatorics Probability and Computing |

Volume | 15 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Jan 2006 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Statistics and Probability

### Cite this

^{2}.

*Combinatorics Probability and Computing*,

*15*(1-2), 213-227. https://doi.org/10.1017/S0963548305007169

**On the number of monochromatic solutions of x+y=z ^{2}.** / Khalfalah, Ayman; Szemerédi, Endre.

Research output: Contribution to journal › Article

^{2}',

*Combinatorics Probability and Computing*, vol. 15, no. 1-2, pp. 213-227. https://doi.org/10.1017/S0963548305007169

^{2}. Combinatorics Probability and Computing. 2006 Jan;15(1-2):213-227. https://doi.org/10.1017/S0963548305007169

}

TY - JOUR

T1 - On the number of monochromatic solutions of x+y=z2

AU - Khalfalah, Ayman

AU - Szemerédi, Endre

PY - 2006/1

Y1 - 2006/1

N2 - In the present work we prove the following conjecture of Erdos, 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)=z2.

AB - In the present work we prove the following conjecture of Erdos, 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)=z2.

UR - http://www.scopus.com/inward/record.url?scp=29744452866&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=29744452866&partnerID=8YFLogxK

U2 - 10.1017/S0963548305007169

DO - 10.1017/S0963548305007169

M3 - Article

VL - 15

SP - 213

EP - 227

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 1-2

ER -