A note on strongly and totally chain intersecting families

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 $A_1⊊A_2⊊\cdots ⊊A_p$ and $q$-chain $B_1⊊B_2⊊\cdots ⊊B_q$, we have $A_p\cap B_q\mathrm{\ne }\mathrm{\varnothing }$. Bernáth and Gerbner determined the largest cardinality of such families. They also introduced strongly $(p,q)$-chain intersecting families, where $A_p\cap B_1\mathrm{\ne }\mathrm{\varnothing }$ and totally $(p,q)$-chain intersecting families, where $A_1\cap B_1\mathrm{\ne }\mathrm{\varnothing }$. 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.