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<p>We study several natural classes of graphs on a zero-dimensional metrizable compact space having no continuous coloring. We compare these graphs with the quasi-order <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_510_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(\preceq ^i_c\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mo>⪯</mo> <mi>c</mi> <mi>i</mi> </msubsup> </math></EquationSource> </InlineEquation> associated with injective continuous homomorphisms. We prove the existence of an antichain basis for these classes. We determine the size of such an antichain basis. We provide a concrete antichain basis when there is a countable one. We also provide some related quasi-orders and equivalence relations which are analytic complete as sets.</p>

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Continuous colorings on compact spaces

  • 期刊论文

  • 发表日期：

    2025年10月1日

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  • Noé de Rancourt,
  • Dominique Lecomte,
  • Miroslav Zelený

摘要

We study several natural classes of graphs on a zero-dimensional metrizable compact space having no continuous coloring. We compare these graphs with the quasi-order \(\preceq ^i_c\) ⪯ c i associated with injective continuous homomorphisms. We prove the existence of an antichain basis for these classes. We determine the size of such an antichain basis. We provide a concrete antichain basis when there is a countable one. We also provide some related quasi-orders and equivalence relations which are analytic complete as sets.