for z$=$2.5. Red circles show the 25 radio galaxies from the present analysis and those of @collet14b, which have measurements of all four lines or sensitive upper limits (on H$\beta$ or \[NII\]$\lambda$6583). We also mark the range of line ratios expected for low-redshift galaxies with ordinary star formation properties and high-metallicity AGN narrow-line regions (light blue hatched region). The red lines show the ’high-redshift’ starburst region of @kewley13a and the AGN branch with low-metallicity narrow-line regions (their scenario 4)](metallicity_hzrg.jpg){width="50.00000%"}

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In Fig. \[fig:metallicity\] we show where our galaxies fall relative to the \[NII\]/H$\alpha$ vs. \[OIII\]/H$\beta$ diagram. Given that our sources host very luminous obscured quasars [e.g., @overzier05; @drouart14], it may not be surprising that our galaxies fall outside the sequence of HII regions, and into the part of the diagram that is characteristic of AGN photoionization [see also @villar97; @villar03; @humphrey09]. However, our galaxies do not fall onto the classical AGN branch from low-redshift studies either, but towards higher \[OIII\]$\lambda$5007 fluxes for a given H$\beta$ flux and \[NII\]/H$\alpha$ ratio.

@kewley13a and @kewley13b predicted such a shift for gas photoionized by powerful AGN, when imposing a redshift evolution of narrow-line metallicity as modeled by @dave10, which scales with the cosmic history 