  To understand how strong emission line galaxies (ELGs) contribute to the
overall growth of galaxies and star formation history of the universe, we
target Strong ELGs (SELGs) from the ZFOURGE imaging survey that have blended
(Hb+[OIII]) rest-frame equivalent widths of >230A and 2.5<zphot<4.0. Using
Keck/MOSFIRE, we measure 49 redshifts for galaxies brighter than Ks=25 mag as
part of our Multi-Object Spectroscopic Emission Line (MOSEL) survey. Our
spectroscopic success rate is ~53% and zphot uncertainty is sigma_z=
[Delta(z)/(1+z)]=0.0135. We confirm 31 ELGs at 3<zspec<3.8 and show that Strong
ELGs have spectroscopic rest-frame [OIII]5007A equivalent widths of 100-500A
and tend to be lower mass systems [log(Mstar/Msun)~8.2-9.6] compared to more
typical star-forming galaxies. The Strong ELGs lie ~0.9 dex above the
star-forming main-sequence at z~3.5 and have high inferred gas fractions of
fgas~>60%, i.e. the inferred gas masses can easily fuel a starburst to double
stellar masses within ~10-100 Myr. Combined with recent results using ZFOURGE,
our analysis indicates that 1) strong [OIII]5007A emission signals an early
episode of intense stellar growth in low mass (Mstar<0.1M*) galaxies and 2)
many, if not most, galaxies at z>3 go through this starburst phase. If true,
low-mass galaxies with strong [OIII]5007A emission (EW_rest>200A) may be an
increasingly important source of ionizing UV radiation at z>3.
