关于单位分数方程的拉多数
On Rado numbers for equations with unit fractions
AI总结：
本研究聚焦单位分数方程的拉多数问题，通过构造满足线性方程单色解且最小公倍数足够小的有限集方法，改进了相关上界，该方法还可推广至其他相关方程的拉多数研究。
摘要：
设$f_r(k)$为满足以下条件的最小正整数$n$：对$\{1,2,...,n\}$的任意$r$着色，非线性方程\\[1/x_1+\cdots+1/x_k=1/y\\]都存在单色解，其中$x_1,...,x_k$未必互不相同。Brown与Rödl[Bull. Aust. Math. Soc. 43(1991): 387-392]证明了$f_2(k)=O(k^6)$。本文证明了$f_2(k)=O(k^3)$。证明的核心要素是一个有限集$A\subseteq\mathbb{N}$，该集合满足：对$A$的任意$2$着色，线性方程$x_1+\cdots+x_k=y$都存在单色解，且$A$的最小公倍数足够小。该方法还可用于研究带有$r>2$的$f_r(k)$。例如，Boza、Marín、Revuelta与Sanz的近期结果[Discrete Appl. Math. 263(2019): 59-68]表明$f_3(k)=O(k^{43})$。
英文摘要：
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 $f_3(k)=O(k^{43})$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
