  It is known that in any $r$-coloring of the edges of a complete $r$-uniform
hypergraph, there exists a spanning monochromatic component. Given a Steiner
triple system on $n$ vertices, what is the largest monochromatic component one
can guarantee in an arbitrary 3-coloring of the edges?
  Gy\'arf\'as proved that $(2n+3)/3$ is an absolute lower bound and that this
lower bound is best possible for infinitely many $n$. On the other hand, we
prove that for almost all Steiner triple systems the lower bound is actually
$(1-o(1))n$. We obtain this result as a consequence of a more general theorem
which shows that the lower bound depends on the size of a largest
\emph{3-partite hole} (that is, sets $X_1, X_2, X_3$ with $|X_1|=|X_2|=|X_3|$
such that no edge intersects all of $X_1, X_2, X_3$) in the Steiner triple
system (Gy\'arf\'as previously observed that the upper bound depends on this
parameter). Furthermore, we show that this lower bound is tight unless the
coloring has a particular structure.
  We also suggest a variety of other Ramsey problems in the setting of Steiner
triple systems.
