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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?\n  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 $(1-o(1))n$. 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