  We evaluate the effective interactions in a fluid of electrons moving in a
plane, on the approach to the quantum phase transition from the paramagnetic to
the fully spin-polarized phase that has been reported from Quantum Monte Carlo
runs. We use the approach of Kukkonen and Overhauser to treat exchange and
correlations under close constraints imposed by sum rules. We show that, as the
paramagnetic fluid approaches the phase transition, the effective interactions
at low momenta develop an attractive region between parallel-spin electrons and
a corresponding repulsive region for antiparallel-spin electron pairs. A
connection with the Hubbard model is made and used to estimate the magnetic
energy gap and hence the temperature at which the phase transition may become
observable with varying electron density in a semiconductor quantum well.
