Bernáth and Gerbner in 2007 introduced (p, q)-chain intersecting families of subsets of an n-element underlying set. Those have the property that for any p-chain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$A_1\subsetneq A_2\subsetneq \dots \subsetneq A_p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mn>1</mn> </msub> <mo>⊊</mo> <msub> <mi>A</mi> <mn>2</mn> </msub> <mo>⊊</mo> <mo>⋯</mo> <mo>⊊</mo> <msub> <mi>A</mi> <mi>p</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and q-chain <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$B_1\subsetneq B_2\subsetneq \dots \subsetneq B_q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mn>1</mn> </msub> <mo>⊊</mo> <msub> <mi>B</mi> <mn>2</mn> </msub> <mo>⊊</mo> <mo>⋯</mo> <mo>⊊</mo> <msub> <mi>B</mi> <mi>q</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, we have <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$A_p\cap B_q\ne \emptyset $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>p</mi> </msub> <mo>∩</mo> <msub> <mi>B</mi> <mi>q</mi> </msub> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mrow> </math></EquationSource> </InlineEquation>. Bernáth and Gerbner determined the largest cardinality of such families. They also introduced strongly (p, q)-chain intersecting families, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A_p\cap B_1\ne \emptyset $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>p</mi> </msub> <mo>∩</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mrow> </math></EquationSource> </InlineEquation> and totally (p, q)-chain intersecting families, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$A_1\cap B_1\ne \emptyset $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mn>1</mn> </msub> <mo>∩</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mrow> </math></EquationSource> </InlineEquation>. They obtained some partial results on the maximum cardinality of such families. We extend those results by determining the largest cardinality of strongly (p, q)-chain intersecting families if n is sufficiently large, and by determining the largest cardinality of totally (2, 2)-chain intersecting families.

A note on strongly and totally chain intersecting families

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2025年10月30日

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Dániel Gerbner

摘要

Bernáth and Gerbner in 2007 introduced (p, q)-chain intersecting families of subsets of an n-element underlying set. Those have the property that for any p-chain $A_1\subsetneq A_2\subsetneq \dots \subsetneq A_p$ $A_{1} ⊊ A_{2} ⊊ \dots ⊊ A_{p}$ and q-chain $B_1\subsetneq B_2\subsetneq \dots \subsetneq B_q$ $B_{1} ⊊ B_{2} ⊊ \dots ⊊ B_{q}$ , we have $A_p\cap B_q\ne \emptyset $ $A_{p} \cap B_{q} \neq \emptyset$ . Bernáth and Gerbner determined the largest cardinality of such families. They also introduced strongly (p, q)-chain intersecting families, where $A_p\cap B_1\ne \emptyset $ $A_{p} \cap B_{1} \neq \emptyset$ and totally (p, q)-chain intersecting families, where $A_1\cap B_1\ne \emptyset $ $A_{1} \cap B_{1} \neq \emptyset$ . They obtained some partial results on the maximum cardinality of such families. We extend those results by determining the largest cardinality of strongly (p, q)-chain intersecting families if n is sufficiently large, and by determining the largest cardinality of totally (2, 2)-chain intersecting families.