An r-coloring of a set S is a function from S to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\{0,1, \cdots , r-1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>r</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(E_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the equation <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$x_1+x_2+\cdots +x_n=y^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>=</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. The r-color Rado number <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$R_r(E_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>r</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(E_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is the least integer R, provided it exists, such that every r-coloring of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\{1,2,\cdots , R\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>R</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> admits a monochromatic solution to <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$(E_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In this study, for each <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$n,\ r\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>,</mo> <mspace width="4pt" /> <mi>r</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, we provide a lower bound <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\ell _r$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$R_r(E_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>r</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Accordingly, we show that when <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$2\le n\le 6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$R_2(E_n)=n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>, and when <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$n\ge 7$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>7</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$R_2(E_n)=\Big \lceil \sqrt{n\lceil { {\sqrt{n}}} \rceil }\Big \rceil $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">⌈</mo> </mrow> <msqrt> <mrow> <mi>n</mi> <mo>⌈</mo> <msqrt> <mi>n</mi> </msqrt> <mo>⌉</mo> </mrow> </msqrt> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">⌉</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which coincides with the lower bound <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\ell _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> that we have provided. We also show that <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$R_3(E_n)=n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$2\le n\le 11$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>11</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$R_3(E_{12})=11$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mn>12</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>11</mn> </mrow> </math></EquationSource> </InlineEquation>.

Two and Three-color Rado numbers for $x_1+x_2+\cdots +x_n=y^2$

期刊论文
发表日期：

2026年1月5日

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Byeong Moon Kim,
Byung Chul Song,
Woonjae Hwang

摘要

An r-coloring of a set S is a function from S to $\{0,1, \cdots , r-1\}$ ${0, 1, \dots, r - 1}$ . Let $(E_n)$ $(E_{n})$ be the equation $x_1+x_2+\cdots +x_n=y^2$ $x_{1} + x_{2} + \dots + x_{n} = y^{2}$ . The r-color Rado number $R_r(E_n)$ $R_{r} (E_{n})$ of $(E_n)$ $(E_{n})$ is the least integer R, provided it exists, such that every r-coloring of $\{1,2,\cdots , R\}$ ${1, 2, \dots, R}$ admits a monochromatic solution to $(E_n)$ $(E_{n})$ . In this study, for each $n,\ r\ge 2$ $n, r \geq 2$ , we provide a lower bound $\ell _r$ $ℓ_{r}$ of $R_r(E_n)$ $R_{r} (E_{n})$ . Accordingly, we show that when $2\le n\le 6$ $2 \leq n \leq 6$ , $R_2(E_n)=n$ $R_{2} (E_{n}) = n$ , and when $n\ge 7$ $n \geq 7$ , $R_2(E_n)=\Big \lceil \sqrt{n\lceil { {\sqrt{n}}} \rceil }\Big \rceil $ $R_{2} (E_{n}) = ⌈ \sqrt{n ⌈ \sqrt{n} ⌉} ⌉$ , which coincides with the lower bound $\ell _2$ $ℓ_{2}$ that we have provided. We also show that $R_3(E_n)=n$ $R_{3} (E_{n}) = n$ when $2\le n\le 11$ $2 \leq n \leq 11$ , and $R_3(E_{12})=11$ $R_{3} (E_{12}) = 11$ .

Two and Three-color Rado numbers for \(x_1+x_2+\cdots +x_n=y^2\)

摘要