  Recently a new class of composite Higgs models have been developed which give
rise to naturally light Higgs bosons without supersymmetry. Based on the chiral
symmetries of ``theory space,'' involving replicated gauge groups and
appropriate gauge symmetry breaking patterns, these models allow the scale of
the underlying strong dynamics giving rise to the composite particles to be as
large as of order 10 TeV, without any fine tuning to prevent large corrections
to Higgs boson mass(es) of order 100 GeV. In this note we show that the size of
flavor violating interactions arising generically from underlying flavor
dynamics constrain the scale of the Higgs boson compositeness to be greater
than of order 75 TeV, implying that significant fine-tuning is required.
Without fine-tuning, the low-energy structure of the composite Higgs model
alone is not sufficient to eliminate potential problems with flavor-changing
neutral currents or excessive CP violation; solving those problems requires
additional information or assumptions about the symmetries of the underlying
flavor or strong dynamics. We also consider the weaker, but more
model-independent, bounds which arise from limits on weak isospin violation.
