  We present a systematic study for ionization state of inter-stellar medium in
galaxies at z=0-3 with ~140,000 SDSS galaxies and 108 intermediate to high
redshift galaxies from the literature, using an ionization-parameter sensitive
line ratio of [OIII]5007/[OII]3727 and photoionization models. We confirm that
z=2-3 galaxies show an [OIII]/[OII] ratio significantly higher than a typical
star-forming galaxy of SDSS by a factor of >~10, and the photoionization models
reveal that these high-z galaxies have an ionization parameter of log(qion/cm
s^{-1})~7.6-9.0, a factor of ~4-10 higher than local galaxies. For galaxies at
any redshift, we identify a correlation between the [OIII]/[OII] ratio and
galaxy global properties of star-formation rate (SFR), stellar mass (M_star),
and metallicity (Z). We extend the fundamental metallicity relation (FMR) and
develop the fundamental ionization relation (FIR), a four-dimensional relation
of ionization parameter, SFR, M_star, and Z. The intermediate and high-z
galaxies up to z~3 follow the FIR defined with the local galaxies, in contrast
with the FMR whose possible evolution from z~2 to 3 is reported. We find that
the FMR evolution of z~2-3 appears, if one omits ionization parameter
differences, and that the FMR evolution does not exist for an average
metallicity solution of z~3 galaxies with a high-ionization parameter.
Interestingly, all of two local Lyman-continuum emitting galaxies (LyC leakers)
have a high [OIII]/[OII] ratio, indicating a positive correlation between
[OIII]/[OII] and ionizing photon escape fraction (fesc), which is successfully
explained by our photoionization models. Because [OIII]/[OII] ratios of z~2-3
galaxies, especially Ly-alpha emitters (LAEs), are comparable to, or higher
than, those of the local LyC leakers, these high-z galaxies are candidates of
Lyman-continuum emitting objects. (abridged)
