On Rado numbers for equations with unit fractions

Abstract: Let $R_r(k)$ be the smallest $n$ such that every $r$-coloring of $\{1,2,...,n\}$ has a monochromatic solution to $x_1+x_2+\cdots+x_k=y$, where $x_1,x_2,\ldots,x_k$ are not necessarily distinct. Beutelspacher and Brestovansky proved that $R_2(k)=k^2+k-1$ and, recently, Boza, Mar\'{i}n, Revuelta, and Sanz proved that $R_3(k)=k^3+2k^2-2$. Similarly, let $f_r(k)$ be the smallest $n$ such that every $r$-coloring of $\{1,2,...,n\}$ has a monochromatic solution to the equation $1/x_1+1/x_2+\cdots+1/x_k=1/y$, where $x_1,x_2,\ldots,x_k$ are not necessarily distinct. Brown and R\"{o}dl proved that $f_2(k)=O(k^6)$. In this talk, we show that $f_2(k)=O(k^3)$ and $f_3(k)=O(k^{43})$. The main ingredient in our proof is a finite set $A\subseteq\mathbb{N}$ such that every $r$-coloring of $A$ has a monochromatic solution to the linear equation $x_1+x_2+\cdots+x_k=y$ and the least common multiple of $A$ is sufficiently small. As for the lower bound, we show that $f_r(k)\geq k^r$ which leads to an interesting open question: Is $f_2(k)=\Theta(k^2)$?

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)