  We consider two functions $\phi$ and $\psi$, defined as follows. Let $x,y \in
(0,1]$ and let $A,B,C$ be disjoint nonempty subsets of a graph $G$, where every
vertex in $A$ has at least $x|B|$ neighbors in $B$, and every vertex in $B$ has
at least $y|C|$ neighbors in $C$. We denote by $\phi(x,y)$ the maximum $z$ such
that, in all such graphs $G$, there is a vertex $v \in C$ that is joined to at
least $z|A|$ vertices in $A$ by two-edge paths. If in addition we require that
every vertex in $B$ has at least $x|A|$ neighbors in $A$, and every vertex in
$C$ has at least $y|B|$ neighbors in $C$, we denote by $\psi(x,y)$ the maximum
$z$ such that, in all such graphs $G$, there is a vertex $v \in C$ that is
joined to at least $z|A|$ vertices in $A$ by two-edge paths. In their recent
paper, M. Chudnovsky, P. Hompe, A. Scott, P. Seymour, and S. Spirkl introduced
these functions, proved some general results about them, and analyzed when they
are greater than or equal to $1/2, 2/3,$ and $1/3$. Here, we extend their
results by analyzing when they are greater than or equal to $3/4, 2/5,$ and
$3/5$.
