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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2306.04029 |
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| _version_ | 1866916299461885952 |
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| author | Gaiser, Collier |
| author_facet | Gaiser, Collier |
| contents | Let $f_r(k)$ be the smallest positive integer $n$ such that every $r$-coloring of $\{1,2,...,n\}$ has a monochromatic solution to the nonlinear equation \[1/x_1+\cdots+1/x_k=1/y,\] where $x_1,...,x_k$ are not necessarily distinct. Brown and Rödl [Bull. Aust. Math. Soc. 43(1991): 387-392] proved that $f_2(k)=O(k^6)$. In this paper, we prove that $f_2(k)=O(k^3)$. The main ingredient in our proof is a finite set $A\subseteq\mathbb{N}$ such that every $2$-coloring of $A$ has a monochromatic solution to the linear equation $x_1+\cdots+x_k=y$ and the least common multiple of $A$ is sufficiently small. This approach can also be used to study $f_r(k)$ with $r>2$. For example, a recent result of Boza, Marín, Revuelta, and Sanz [Discrete Appl. Math. 263(2019): 59-68] implies that $f_3(k)=O(k^{43})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_04029 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Rado numbers for equations with unit fractions Gaiser, Collier Combinatorics Let $f_r(k)$ be the smallest positive integer $n$ such that every $r$-coloring of $\{1,2,...,n\}$ has a monochromatic solution to the nonlinear equation \[1/x_1+\cdots+1/x_k=1/y,\] where $x_1,...,x_k$ are not necessarily distinct. Brown and Rödl [Bull. Aust. Math. Soc. 43(1991): 387-392] proved that $f_2(k)=O(k^6)$. In this paper, we prove that $f_2(k)=O(k^3)$. The main ingredient in our proof is a finite set $A\subseteq\mathbb{N}$ such that every $2$-coloring of $A$ has a monochromatic solution to the linear equation $x_1+\cdots+x_k=y$ and the least common multiple of $A$ is sufficiently small. This approach can also be used to study $f_r(k)$ with $r>2$. For example, a recent result of Boza, Marín, Revuelta, and Sanz [Discrete Appl. Math. 263(2019): 59-68] implies that $f_3(k)=O(k^{43})$. |
| title | On Rado numbers for equations with unit fractions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2306.04029 |