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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathcal {H}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> be a <i>k</i>-uniform hypergraph on <i>n</i> vertices. For a fixed positive integer <i>s</i>, the most natural way to define a subhypergraph of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathcal {H}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> without <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(s + 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> pairwise disjoint edges is to take all edges containing at least one of <i>s</i> fixed vertices. We prove that for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n &gt; n_0(k, s)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>&gt;</mo> <msub> <mi>n</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and a relatively wide class of hypergraphs this construction is the largest.</p>

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On the maximum size of substructures with fixed matching number

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月25日

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  • P. Frankl

摘要

Let \({\mathcal {H}}\) H be a k-uniform hypergraph on n vertices. For a fixed positive integer s, the most natural way to define a subhypergraph of \({\mathcal {H}}\) H without \(s + 1\) s + 1 pairwise disjoint edges is to take all edges containing at least one of s fixed vertices. We prove that for \(n > n_0(k, s)\) n > n 0 ( k , s ) and a relatively wide class of hypergraphs this construction is the largest.