  We address several extremal problems concerning the spreading property of
point sets of Steiner triple systems. This property is closely related to the
structure of subsystems, as a set is spreading if and only if there is no
proper subsystem which contains it. We give sharp upper bounds on the size of a
minimal spreading set in a Steiner triple system and show that if all the
minimal spreading sets are large then the examined triple system must be a
projective space. We also show that the size of a minimal spreading set is not
an invariant of a Steiner triple system.
