A note on strongly and totally chain intersecting families

Abstract 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 $$A_1\subsetneq A_2\subsetneq \dots \subsetneq A_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>⊊</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>⊊</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>⊊</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:math> and q -chain $$B_1\subsetneq B_2\subsetneq \dots \subsetneq B_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>⊊</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>⊊</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>⊊</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:mrow> </mml:math> , we have $$A_p\cap B_q\ne \emptyset $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>∩</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo>≠</mml:mo> <mml:mi>∅</mml:mi> </mml:mrow> </mml:math> . Bernáth and Gerbner determined the largest cardinality of such families. They also introduced strongly ( p , q )-chain intersecting families, where $$A_p\cap B_1\ne \emptyset $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>∩</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>≠</mml:mo> <mml:mi>∅</mml:mi> </mml:mrow> </mml:math> and totally ( p , q )-chain intersecting families, where $$A_1\cap B_1\ne \emptyset $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>∩</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>≠</mml:mo> <mml:mi>∅</mml:mi> </mml:mrow> </mml:math> . 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.