  We present the results of the first X-ray all-sky survey (eRASS1) performed
by the eROSITA instrument onboard the Spectrum-Roentgen-Gamma (SRG) mission on
X-ray emitting red giants and supergiants. Focussing on stars positioned at
high galactic latitudes above 20 deg, we construct a complete sample of such
objects using the Gaia DR3 catalog and identify a sample 96 stars appearing as
bona fide entries in the eRASS1 source catalog. Restricting again the sample to
objects nearer than 1300~pc and eliminating all catalog entries which are due
to optical contamination, we end up with a sample of 16 genuine red
giant/supergiant X-ray sources, which represent -- with the exception of one
source (CL~Hyi) -- new X-ray detections. We furthermore present a low SNR X-ray
spectrum of the nearby low activity giant Arcturus obtained from a pointed
observation with the XMM-Newton satellite and give a detailed account of our
data analysis. We show that Arcturus-like X-ray emission cannot be the
explanation for the X-ray emissions observed by eROSITA and provide a
discussion of the possible nature of the detected X-ray sources.
