What is the temperature structure in the giant HII region NGC 588?

We present the results of an exhaustive study of the ionized gas in NGC 588, a giant H II region in the nearby spiral galaxy M33. This analysis uses a high number of diagnostics in the optical and infrared ranges. Four temperature diagnostics obtained with optical lines agree with a gas temperature of 11000K, while the [O III] 5007/88um ratio yields a much lower temperature of 8000K approximately. This discrepancy suggests the presence of large temperature inhomogeneities in the nebula. We investigated the cause of this discrepancy by constructing photoionization models of increasing complexity. In particular, we used the constraints from the Halpha and Hbeta surface brightness distributions and state-of-the-art models of the stellar ionizing spectrum. None of the successive attempts was able to reproduce the discrepancy between the temperature diagnostics, so the thermal balance of NGC 588 remains unexplained. We give an estimate of the effect of this failure on the O/H and Ne/O estimates and show that O/H is known to within +/-0.2 dex