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A sharp lower bound for some reciprocal Rado numbers

This paper establishes sharp lower bounds for reciprocal Rado numbers $f_r(k)$ , proving specific inequalities for general $r$ and $k$ , determining exact values or improved bounds for the case $r=2$ under certain conditions on $k$ , and providing new computational results and generalizations.

Original authors: Collier Gaiser, Mojtaba Ramezanpour

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Collier Gaiser, Mojtaba Ramezanpour

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant bag of numbered tiles, from 1 up to some huge number $n$ . You also have a set of crayons (colors). Your goal is to color every single tile in the bag.

The paper explores a specific mathematical game involving these colored tiles and a very specific rule about "unit fractions" (fractions where the top number is always 1, like 1/2, 1/3, 1/4).

The Game: The "Reciprocal Rado" Challenge

The rule of the game is this equation:
$\frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_k} = \frac{1}{x_{k+1}}$

Think of this as a recipe. You need to pick $k$ numbers (let's call them ingredients) from your bag. When you add up their "reciprocals" (flip them upside down), they must perfectly equal the reciprocal of one final number (the "dish").

The Catch: All the numbers you pick for the ingredients and the dish must be the same color.

The big question the authors ask is: How big does your bag of tiles ( $n$ ) need to be before you are guaranteed to find a solution, no matter how you try to color the tiles to avoid it?

This minimum number is called a Reciprocal Rado number, denoted as $f_r(k)$ .

$r$ is the number of colors you have.
$k$ is the number of ingredients in your recipe.

The Authors' Discovery: "The Safety Net"

The authors wanted to find a "sharp lower bound." In plain English, they wanted to prove a "safety net" size. They wanted to say, "If your bag is smaller than this specific number, you can always find a way to color the tiles so that no single-color recipe works."

They proved two main things:

For 2 ingredients ( $k=2$ ): If you have $r$ colors, your bag needs to be at least roughly $4^{r/2}$ big to force a solution.
- Analogy: Imagine you are trying to hide a specific pair of matching socks in a drawer. The authors calculated exactly how many socks you need to have in the drawer before it becomes mathematically impossible to hide the pair without them being the same color.
For 3 or more ingredients ( $k \ge 3$ ): If you have $r$ colors, the bag needs to be at least $(2r - 1) \times k^r$ big.
- Analogy: As the recipe gets more complex (more ingredients), the "hiding spot" gets harder to find, but the authors found a formula that guarantees the hiding spot disappears once the bag gets big enough.

The Special Case: Two Colors ( $r=2$ )

The paper dives deep into the scenario where you only have Red and Blue crayons. This is the most common version of the game.

They found a very precise pattern for when the "safety net" is tight (meaning the number is exactly what the formula predicts) and when it's loose (meaning you need a bigger bag than the formula suggests).

The "Perfect" Matches: If the number of ingredients $k$ $k$ is a multiple of 3 times a power of 2 (like 6, 12, 24), the magic number is exactly $3k^2$ $3 k^{2}$ .
- Example: If you need 6 ingredients, the bag size is exactly $3 \times 6^2 = 108$ .
The "Odd Prime" Twist: If the number of ingredients $k$ $k$ is an odd prime number raised to a power (like 3, 5, 7, 9, 11, 25), the magic number is slightly higher: at least $3k^2 + 1$ $3 k^{2} + 1$ .
- Analogy: It's like a puzzle where most pieces fit perfectly into a box, but if the piece is shaped like a "prime number," it sticks out just a tiny bit, requiring a slightly larger box.

The Computer Detective Work

The authors didn't just do math on paper; they used computers to act as "detectives." They wrote code to test every possible way to color bags of numbers up to size 25.

They confirmed that for numbers like 6, 8, 10, 12, etc., the formula $3k^2$ works perfectly.
They confirmed that for odd prime powers like 5, 7, 9, 11, the formula needs that extra "+1" bump.
They even found that a previous study had a mistake (claiming a number was 39 when it had to be at least 75), and they corrected it.

Based on their computer results, they made a Conjecture (a strong guess):

"If $k$ is not an odd prime power (and is at least 4), then the magic number is exactly $3k^2$ ."

The Generalized Recipe

Finally, the authors looked at a more complex version of the game where the ingredients have "weights" (coefficients). Instead of just adding $1/x$ , you might add $2/x$ or $5/x$ .

They proved that even with these weights, you can still calculate a "safety net" size. The size depends on the smallest weight and the sum of all weights. It's like saying, "Even if your recipe calls for heavy spices, we can still tell you exactly how big your pantry needs to be to guarantee you'll find a matching set of colored jars."

Summary

This paper is about finding the exact tipping point where chaos (random coloring) turns into order (a guaranteed monochromatic solution) for a specific type of fraction equation. The authors provided new, tighter formulas for this tipping point, used computers to verify them for small numbers, and discovered a special rule that applies when the number of ingredients is an "odd prime power."

Technical Summary: A Sharp Lower Bound for Some Reciprocal Rado Numbers

Problem Statement
This paper addresses a problem within arithmetic Ramsey theory concerning "reciprocal Rado numbers." Specifically, it investigates the equation involving unit fractions:
$\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_k} = \frac{1}{x_{k+1}}$
where $x_1, \dots, x_k$ are not necessarily distinct positive integers. Let $f_r(k)$ denote the smallest positive integer $n$ such that every $r$ -coloring of the set $\{1, 2, \dots, n\}$ contains a monochromatic solution to this equation. While Brown and Rödler [4] and Lefmann [11] established that such an $n$ exists for sufficiently large values, precise bounds and exact values for $f_r(k)$ remain largely unknown. The paper aims to improve existing lower bounds for $f_r(k)$ and determine exact values or tighter bounds for specific cases, particularly when $r=2$ .

Methodology
The authors employ a combination of combinatorial construction, number-theoretic arguments, and computational verification:

Constructive Lower Bounds: To establish lower bounds for $f_r(k)$ , the authors construct explicit $r$ -colorings of intervals $\{1, 2, \dots, N\}$ that avoid monochromatic solutions. These constructions partition the integer set into disjoint intervals ( $A_1, A'_i, A''_i$ ) assigned specific colors. The core of the proof relies on analyzing the magnitude of the terms in the reciprocal equation. A key tool is Lemma 2.1, which provides a lower bound for the difference between reciprocals of distinct integers within a specific range, used to derive contradictions when assuming a solution exists within a single color class.
Case Analysis for $r=2$ : For the 2-coloring case, the authors analyze specific forms of $k$ . They utilize induction and specific algebraic identities involving powers of 2 and multiples of $k$ to demonstrate that certain colorings of $\{1, \dots, 3k^2\}$ fail to produce monochromatic solutions under specific conditions (e.g., when $k$ is an odd prime power).
Computational Verification: The authors utilized Python scripts assisted by AI tools (ChatGPT 5.5 and ChatGPT Codex) and Boolean Satisfiability (SAT) solvers (Cadical195 and Glucose3) to compute exact values of $f_2(k)$ and $f_3(k)$ for ranges of $k$ not previously determined. These computations were performed on an AMD Ryzen 7 9800X3D processor.

Key Contributions and Results

Improved General Lower Bounds:
The paper proves two primary theorems improving upon previous bounds (such as $f_r(k) \ge k^r$ ):
- For $r \ge 1$ , $f_r(2) \ge 4^{r/2}$ .
- For $k \ge 3$ and $r \ge 1$ , $f_r(k) \ge (2r - 1)k^r$ .
  The authors note that for $r=2$ , the bound $f_2(k) \ge 3k^2$ is sharp for infinitely many $k$ .
Exact Values and Tighter Bounds for $r=2$ :
- Case $k = 3 \cdot 2^m$ : The authors prove that if $k$ is of the form $3 \cdot 2^m$ (for positive integer $m$ ), then $f_2(k) = 3k^2$ .
- Case $k = p^m$ (Odd Prime Power): If $k$ is an odd prime power, the authors prove $f_2(k) \ge 3k^2 + 1$ .
- Computational Results: The paper provides a table of computed values for $f_2(k)$ for $k \in \{2, \dots, 25\}$ . These results correct a previous claim by Myers and Parrish [12] regarding $f_2(5)$ , showing it to be 80 (consistent with the new lower bound) rather than 39. The data suggests that $f_2(k) = 3k^2$ if and only if $k$ is not an odd prime power (for $k \ge 4$ ).
Generalization to Coefficients:
The authors generalize their lower bound results to equations with arbitrary positive integer coefficients:
$\frac{a_1}{x_1} + \frac{a_2}{x_2} + \cdots + \frac{a_k}{x_k} = \frac{1}{x_{k+1}}$
They establish that for 2-colorings, $f(a_1, a_2) \ge 2\min\{a_1, a_2\}(a_1+a_2)^2$ and for $k \ge 3$ , $f(a_1, \dots, a_k) \ge (2\min\{a_i\} + 1)(\sum a_i)^2$ .
Computational Data for $r=3$ :
New computational results for $f_3(k)$ are provided for $k \in \{2, 3, 4, 5\}$ . The authors observe that the lower bound $f_3(k) \ge 7k^3$ (derived from their general theorem) appears to be loose, as the computed values exceed this bound significantly.

Significance and Claims
The paper claims to provide the first sharp lower bounds for reciprocal Rado numbers in the general case and identifies specific infinite families of $k$ where the lower bound for $f_2(k)$ is exact. The authors highlight that their work refines the understanding of the threshold at which monochromatic solutions to reciprocal equations become unavoidable.

The authors modestly frame their findings by:

Proposing Conjecture 1.3: That for $k \ge 4$ , $f_2(k) = 3k^2$ if and only if $k$ is not an odd prime power.
Identifying open problems, including improving the lower bound for $f_3(k)$ , refining the bound for $f_2(k)$ when $k$ is an odd prime power, and determining if a quadratic upper bound exists for $f_2(k)$ for all $k \ge 4$ .
Noting that while their lower bounds are sharp for specific cases, the general behavior for $r \ge 3$ remains an open area of investigation, as suggested by the discrepancy between their theoretical lower bound and computational results for $f_3(k)$ .

The paper concludes by acknowledging the utility of AI-assisted coding in their computational efforts and inviting further research into the minimum distance properties of unit fractions (Problem 5.4).