tokens: agreement with the numbers published by Hassan et al. ( 2017 ) (∼50 sources in 5 h and in 10 000 deg 2 ), while it is intermediate (but broadly consistent, considering the large uncertainties) between the expectations discussed by Padovani Giommi ( 2015 ) (100–160 BL Lacs expected) and the values estimated by De Franco et al. ( 2017 ) (24 detections). The predictions by Hassan et al. ( 2017 ) are extrapolated by the Fermi -LAT (3FHL) catalogue and, therefore, they consider only the blazars currently detected by Fermi -LAT. In our work, we are considering also Fermi -LAT undetected objects but, as discussed above, the expected detections in 5 h exposure times (3 objects) are all present in current Fermi -LAT catalogue (FL8Y). This means that current predictions for a (shallow) CTA extragalactic survey based on Fermi -LAT catalogue should be reliable while, for deeper pointing, the current Fermi -LAT catalogues could miss up to 50 per cent of objects. In the very close future e-ROSITA (extended ROentgen Survey with an Imaging Telescope Array, Merloni et al. 2012 ) will survey the entire sky, reaching in the soft X-ray band (0.5–2 keV) a flux 100 times fainter than the ROSAT All Sky SURVEY (RASS) data and with an on-axis spatial resolution of ∼15 arcsec. A cross-match of the point sources detected by e-ROSITA with radio catalogues as the NVSS or the Sydney University Molonglo Sky Survey (SUMSS) will provide virtually all the expected ∼800 HBL that could be detected by CTA. 8 SUMMARY AND CONCLUSIONS We have presented a well-defined and representative sample of HBL selected from the REX survey that was obtained by cross-matching radio (NVSS) and X-ray data ( ROSAT -PSPC pointed data). In particular, we selected all the REX with an X-ray-to-radio flux ratio typical of an HBL and falling in a well-defined area of sky covering about 560 deg 2 . We then completed the spectroscopic follow-up of all these sources and we focused on the 46 sources showing no emission lines in the optical spectrum. We called these source Te-REX, standing for TeV-emitting REX candidates, since HBL are the most promising sources to be detected at TeV energies. Out of these 46 sources, 25 have a clear non-thermal nucleus detected in the optical spectrum as a significant reduction of the Ca ii break (40 per cent). In the remaining 21 Te-REX sources, instead, we measured a Ca ii break value above 40 per cent, i.e. the non-thermal nucleus is swamped by the stellar light. We called these 21 sources PEGs (for ‘PEGs’). We expect that most PEGs should be faint HBL nuclei dominated by the host galaxy light. We then evaluated the intensity of the VHE emission of all the 46 Te-REX. To this end, we compare the predicted SEDs with the sensitivity curves of current and future Cherenkov telescopes. Considering deep, pointed observations of 50 h, we found 14 Te-REX (2.5 objects per 100 deg 2 , corresponding to ∼800 objects on the entire extragalactic sky above ∣ b ∣ ≥ 10°) that could be
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