  Starting from an innocent Ramsey-theoretic question regarding directed paths
in tournaments, we discover a series of rich and surprising connections that
lead into the theory around a fundamental problem in Combinatorics: the
Ruzsa-Szemer\'edi induced matching problem. Using these relationships, we prove
that every coloring of the edges of the transitive $n$-vertex tournament using
three colors contains a directed path of length at least $\sqrt{n} \cdot
e^{\log^* n}$ which entirely avoids some color. We also expose connections to a
family of constructions for Ramsey tournaments, and introduce and resolve some
natural generalizations of the Ruzsa-Szemer\'edi problem which we encounter
through our investigation.
