Do red giant stars pulsate in high overtones?

About 30 photometrically variable red giant stars have periods less than 10 d, as determined by the compilers of the Hipparcos Catalogue from Hipparcos photometric measurements. These periods, when combined with estimates of the radii and masses of these stars, and with pulsation theory, imply that these stars are pulsating in very high overtones. We present several pieces of evidence which suggest that the periods may be spurious, as a result of the particular aliasing properties of the Hipparcos photometry. We conclude that the evidence for high-overtone pulsation in red giant stars is equivocal.