  We prove two new equivalences of the Feichtinger conjecture that involve
reproducing kernel Hilbert spaces. We prove that if for every Hilbert space,
contractively contained in the Hardy space, each Bessel sequence of normalized
kernel functions can be partitioned into finitely many Riesz basic sequences,
then a general bounded Bessel sequence in an arbitrary Hilbert space can be
partitioned into finitely many Riesz basic sequences. In addition, we examine
some of these spaces and prove that for these spaces bounded Bessel sequences
of normalized kernel functions are finite unions of Riesz basic sequences.
