A sharp lower bound for some reciprocal Rado numbers

Abstract

Let $f_r(k)$ be the smallest $n$ such that every $r$-coloring of $\{1,2,\ldots,n\}$ has a monochromatic solution to the equation \[\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_k}=\frac{1}{x_{k+1}}, \] where $x_1,x_2,\ldots,x_k$ are not necessarily distinct. In this paper, we prove that $f_r(2)\geq 4^r/2$ for all $r\geq1$, and $f_r(k)\geq(2^r-1)k^r$ for all $k\geq3$ and $r\geq1$. When $r=2$, we show that, if $k=3\cdot2^m$ for some positive integer $m$, then $f_2(k)=3k^2$; and if $k=p^m$ for some odd prime number $p$ and positive integer $m$, then $f_2(k)\geq3k^2+1$. We also provide new computational results for $f_2(k)$ and $f_3(k)$, as well as a generalization of our lower bounds for $f_2(k)$ to equations with general coefficients.

Disclosure

“(2i − 2) > (2i − 1)2i , which is again a contradiction. 3 2-Colorings In this section, we first present some computational results for f2 (k). Our computation is conducted using Python, and the code is developed with the assistance of ChatGPT 5.5. The Python script is available from the first author’s website. We use Boolean satisfiability (SAT) solvers Cadical195 and Glucose3 in the PySAT toolkit for the computation [9]. The CPU of the computer we use is AMD Ryzen 7 9800X3D. F”

PDF page 6

Classification

Computational experiments or data processing

Multiplier

3

Verified

2026-08-24T00:34:05.733Z

Structural counts