  The possibility of a connection between dark energy and gravity through a
direct coupling in the Lagrangian of the underlying theory has acquired an
increasing interest due to the recently discovered capability of the extended
quintessence model to encompass the fine-tuning problem of the cosmological
constant. The gravity induced "R-boost" mechanism is indeed responsible for an
early, enhanced scalar field dynamics, by virtue of which the residual imprint
of a wide set of initial field values is cancelled out. The initial conditions
problem is particularly relevant, as the most recent observations indicate that
the dark energy equation of state approaches, at the present time, the
cosmological constant value, wDE = -1; if confirmed, such observational
evidence would cancel the advantage of a standard, minimally coupled scalar
field as a Dark Energy candidate instead of the cosmological constant, because
of the huge fine tuning it would require. We give here a general classification
of the scalar-tensor gravity theories admitting R-boost solutions scaling as a
power of the cosmological redshift, outlining those behaving as an attractor
for the quintessence field. In particular, we show that all the R-boost
solutions with the dark energy density scaling as the relativistic matter or
shallower represent attractors. This analysis is exhaustive as for the
classification of the couplings which admit R-boost and the subsequent
enlargement of the basin of attraction enclosing the initial scalar field
values.
