  We present a detailed study of the non-Local Thermodynamic Equilibrium
(non-LTE) formation of the high-excitation neutral oxygen 777 nm triplet in
MARCS model atmospheres representative of late-type stars with spectral types F
to K. We carried out the calculations using the statistical equilibrium code
MULTI, including estimates of the impact on elemental abundance analysis. The
atomic model employed includes, in particular, recent quantum-mechanical
electron collision data. We confirm that the O I triplet lines form under
non-LTE conditions in late-type stars, suffering negative abundance corrections
with respect to LTE. At low metallicity, large line opacity stems from
triplet-quintet intersystem electron collisions, a form of coupling previously
not considered or seriously underestimated. The non-LTE effects become
generally severe for models (both giants and dwarfs) with higher T_eff.
Interestingly, in metal-poor turn-off stars, the negative non-LTE abundance
corrections tend to rapidly become more severe towards lower metallicity. When
neglecting H collisions, they amount to as much as ~ 0.9 dex and ~ 1.2 dex,
respectively at [Fe/H]=-3 and [Fe/H]=-3.5. Even when such collisions are
included, the LTE abundance remains a serious overestimate, correspondingly by
~ 0.5 dex and ~ 0.9 dex at such low metallicities. Although the poorly known
inelastic hydrogen collisions thus remain an important uncertainty, the large
metallicity-dependent non-LTE effects seem to point to a resulting "low"
(compared to LTE) [O/Fe] in metal-poor halo stars.
