Optimal bounds on the polynomial Schur's theorem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913292880969728 |
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| author | Kim, Jaehoon Liu, Hong Pach, Péter Pál |
| author_facet | Kim, Jaehoon Liu, Hong Pach, Péter Pál |
| contents | Liu, Pach and Sándor recently characterized all polynomials $p(z)$ such that the equation $x+y=p(z)$ is $2$-Ramsey, that is, any $2$-coloring of $\mathbb{N}$ contains infinitely many monochromatic solutions for $x+y=p(z)$. In this paper, we find asymptotically tight bounds for the following two quantitative questions.
$\bullet$ For $n\in \mathbb{N}$, what is the longest interval $[n,f(n)]$ of natural numbers which admits a $2$-coloring with no monochromatic solutions of $x+y=p(z)$?
$\bullet$ For $n\in \mathbb{N}$ and a $2$-coloring of the first $n$ integers $[n]$, what is the smallest possible number $g(n)$ of monochromatic solutions of $x+y=p(z)$?
Our theorems determine $f(n)$ up to a multiplicative constant $2+o(1)$, and determine the asymptotics for $g(n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_00794 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal bounds on the polynomial Schur's theorem Kim, Jaehoon Liu, Hong Pach, Péter Pál Combinatorics Liu, Pach and Sándor recently characterized all polynomials $p(z)$ such that the equation $x+y=p(z)$ is $2$-Ramsey, that is, any $2$-coloring of $\mathbb{N}$ contains infinitely many monochromatic solutions for $x+y=p(z)$. In this paper, we find asymptotically tight bounds for the following two quantitative questions. $\bullet$ For $n\in \mathbb{N}$, what is the longest interval $[n,f(n)]$ of natural numbers which admits a $2$-coloring with no monochromatic solutions of $x+y=p(z)$? $\bullet$ For $n\in \mathbb{N}$ and a $2$-coloring of the first $n$ integers $[n]$, what is the smallest possible number $g(n)$ of monochromatic solutions of $x+y=p(z)$? Our theorems determine $f(n)$ up to a multiplicative constant $2+o(1)$, and determine the asymptotics for $g(n)$. |
| title | Optimal bounds on the polynomial Schur's theorem |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2404.00794 |