  We study the relationship between stellar mass, star formation rate
(SFR),ionization state, and gas-phase metallicity for a sample of 41 normal
star-forming galaxies at $3 \lesssim z \lesssim 3.7$. The gas-phase oxygen
abundance, ionization parameter, and electron density of ionized gas are
derived from rest-frame optical strong emission lines measured on near-infrared
spectra obtained with Keck/MOSFIRE. We remove the effect of these strong
emission lines in the broad-band fluxes to compute stellar masses via spectral
energy distribution fitting, while the SFR is derived from the dust-corrected
ultraviolet luminosity. The ionization parameter is weakly correlated with the
specific SFR, but otherwise the ionization parameter and electron density do
not correlate with other global galaxy properties such as stellar mass, SFR,
and metallicity. The mass-metallicity relation (MZR) at $z\simeq3.3$ shows
lower metallicity by $\simeq 0.7$ dex than that at $z=0$ at the same stellar
mass. Our sample shows an offset by $\simeq 0.3$ dex from the locally defined
mass-metallicity-SFR relation, indicating that simply extrapolating such
relation to higher redshift may predict an incorrect evolution of MZR.
Furthermore, within the uncertainties we find no SFR-metallicity correlation,
suggesting a less important role of SFR in controlling the metallicity at high
redshift. We finally investigate the redshift evolution of the MZR by using the
model by Lilly et al. (2013), finding that the observed evolution from $z=0$ to
$z\simeq3.3$ can be accounted for by the model assuming a weak redshift
evolution of the star formation efficiency.
