  Assume that $R_1,R_2,\dots,R_t$ are disjoint parallel lines in the plane. A
$t$-interval (or $t$-track interval) is a set that can be written as the union
of $t$ closed intervals, each on a different line. It is known that pairwise
intersecting $2$-intervals can be pierced by two points, one from each line.
However, it is not true that every set of pairwise intersecting $3$-intervals
can be pierced by three points, one from each line. For $k\ge 3$, Kaiser and
Rabinovich asked whether $k$-wise intersecting $t$-intervals can be pierced by
$t$ points, one from each line. Our main result provides an asymptotic answer:
in any set $S_1,\dots,S_n$ of $k$-wise intersecting $t$-intervals, at least
$\frac{k-1}{k+1}n$ can be pierced by $t$ points, one from each line. We prove
this in a more general form, replacing intervals by subtrees of a tree. This
leads to questions and results on covering vertices of edge-colored complete
graphs by vertices of monochromatic cliques having distinct colors, where the
colorings are chordal, or more generally induced $C_4$-free graphs. For
instance, we show that if the edges of a complete graph $K_n$ are colored with
red or blue so that both color classes are induced $C_4$-free, then at least
${4n\over 5}$ vertices can be covered by a red and a blue clique, and this is
best possible. We conclude by pointing to new Ramsey-type problems emerging
from these restricted colorings.
