  We embark on systematic explorations of the behaviour of tetraquark mesons
(colour-singlet bound states of two quarks and two antiquarks) in the
(idealized) limit of a large number $N_c$ of colour degrees of freedom of
quantum chromodynamics. Considering the scattering of two ordinary mesons into
two ordinary mesons, we start off with formulating a set of selection criteria
that should enable us to unambiguously single out precisely those contributions
to all encountered scattering amplitudes that potentially will develop
tetraquark poles. Assuming that tetraquark mesons do exist and, if so, emerge
in the contributions compatible with our criteria at largest admissible order
of $N_c$, we deduce, for the categories of tetraquarks that exhibit either four
or only two different open quark flavours, that the decay rates of these
tetraquark types are, at least, of order $1/N_c^2$ and that internal
consistency requires all the members of the first species to exist pairwise,
distinguishable by their favoured two-ordinary-meson decay channels.
