Large monochromatic components in 3-edge-colored Steiner triple systems.

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? We show that $(2n+3)/3$ is an absolute lower bound, and we construct an infinite family of Steiner triple systems which shows that this lower bound is asymptotically best possible. On the other hand, we show that for almost all Steiner triple systems the lower bound is actually $(1-o(1))n$. We also suggest a variety of other Ramsey problems in the setting of Steiner triple systems.