<p>We settle the Ramsey problem <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2896_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="106" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {R}}(K_6-e,K_4)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">R</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mn>6</mn> </msub> <mo>-</mo> <mi>e</mi> <mo>,</mo> <msub> <mi>K</mi> <mn>4</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, also known as <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2896_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {R}}(J_6,K_4)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">R</mi> <mo stretchy="false">(</mo> <msub> <mi>J</mi> <mn>6</mn> </msub> <mo>,</mo> <msub> <mi>K</mi> <mn>4</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2896_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {R}}(K_6^-,K_4)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">R</mi> <mo stretchy="false">(</mo> <msubsup> <mi>K</mi> <mn>6</mn> <mo>-</mo> </msubsup> <mo>,</mo> <msub> <mi>K</mi> <mn>4</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Previously, the best bounds were <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2896_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="184" /> </InlineMediaObject> <EquationSource Format="TEX">$30\le {\mathcal {R}}(K_6-e,K_4) \le 32$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>30</mn> <mo>≤</mo> <mi mathvariant="script">R</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mn>6</mn> </msub> <mo>-</mo> <mi>e</mi> <mo>,</mo> <msub> <mi>K</mi> <mn>4</mn> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>32</mn> </mrow> </math></EquationSource> </InlineEquation>. We prove that <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2896_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="144" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {R}}(K_6-e,K_4) =30$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">R</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mn>6</mn> </msub> <mo>-</mo> <mi>e</mi> <mo>,</mo> <msub> <mi>K</mi> <mn>4</mn> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>30</mn> </mrow> </math></EquationSource> </InlineEquation>. Our technique is based on the recent approach of Angeltveit and McKay and on older algorithms of McKay and Radziszowski.</p>

${\mathcal {R}}(K_6-e,K_4) =30 ^*$

期刊论文
发表日期：

2025年1月22日

查看全文

David James,
Elisha Kahan,
Erik Rauer

摘要

We settle the Ramsey problem ${\mathcal {R}}(K_6-e,K_4)$ $R (K_{6} - e, K_{4})$ , also known as ${\mathcal {R}}(J_6,K_4)$ $R (J_{6}, K_{4})$ and ${\mathcal {R}}(K_6^-,K_4)$ $R (K_{6}^{-}, K_{4})$ . Previously, the best bounds were $30\le {\mathcal {R}}(K_6-e,K_4) \le 32$ $30 \leq R (K_{6} - e, K_{4}) \leq 32$ . We prove that ${\mathcal {R}}(K_6-e,K_4) =30$ $R (K_{6} - e, K_{4}) = 30$ . Our technique is based on the recent approach of Angeltveit and McKay and on older algorithms of McKay and Radziszowski.

\({\mathcal {R}}(K_6-e,K_4) =30 ^*\)

摘要