<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2892_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> be a constant and let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2892_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$m\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> be an integer. In this short note, we shall show that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2892_Article_IEq3.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="291" /> </InlineMediaObject> <EquationSource Format="TEX">$R(K_{m,\alpha n},K_{m,n})=((\alpha ^{1/m}+1)^m+o(1))n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>α</mi> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>K</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>m</mi> </msup> <mo>+</mo> <mi>o</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2892_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$n\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

Ramsey Numbers of Complete Bipartite Graphs

期刊论文
发表日期：

2025年1月23日

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Meng Liu,
Bangwei Du

摘要

Let $\alpha >0$ $α > 0$ be a constant and let $m\ge 1$ $m \geq 1$ be an integer. In this short note, we shall show that $R(K_{m,\alpha n},K_{m,n})=((\alpha ^{1/m}+1)^m+o(1))n$ $R (K_{m, α n}, K_{m, n}) = ({(α^{1 / m} + 1)}^{m} + o (1)) n$ as $n\rightarrow \infty $ $n \to \infty$ .