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<p>It is proved that the inequality <Equation ID="Equ16"> <EquationSource Format="TEX">\( \chi (\mathbb {R}^3 \times [0,\varepsilon ]^6) \ge 10, \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>χ</mi> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo>×</mo> <msup> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>ε</mi> <mo stretchy="false">]</mo> </mrow> <mn>6</mn> </msup> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>10</mn> <mo>,</mo> </mrow> </math></EquationSource> </Equation>holds true for an arbitrary <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varepsilon &gt; 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\chi (M)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is the chromatic number of an (infinite) graph with vertex set <i>M</i>, and in which two vertices are adjacent if they are at the distance 1. Bibliography: 15 titles.</p>

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ON CHROMATIC NUMBERS OF 3-DIMENSIONAL SLICES

  • 期刊论文

  • 发表日期：

    2026年3月17日

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  • D. D. Cherkashin,
  • A. J. Kanel-Belov,
  • G. A. Strukov,
  • V. A. Voronov

摘要

It is proved that the inequality \( \chi (\mathbb {R}^3 \times [0,\varepsilon ]^6) \ge 10, \) χ ( R 3 × [ 0 , ε ] 6 ) ≥ 10 , holds true for an arbitrary \(\varepsilon > 0\) ε > 0 , where \(\chi (M)\) χ ( M ) is the chromatic number of an (infinite) graph with vertex set M, and in which two vertices are adjacent if they are at the distance 1. Bibliography: 15 titles.