We have conducted a near-infrared monitoring campaign at the UK InfraRed Telescope (UKIRT), of the Local Group spiral galaxy M33 (Triangulum). The main aim was to identify stars in the very final stage of their evolution, and for which the luminosity is more directly related to the birth mass than the more numerous less-evolved giant stars that continue to increase in luminosity. In this fourth paper of the series, we present a search for variable red giant stars in an almost square degree region comprising most of the galaxys disc, carried out with the WFCAM instrument in the K band. These data, taken during the period 2005--2007, were complemented by J- and H-band images. Photometry was obtained for 403 734 stars in this region; of these, 4643 stars were found to be variable, most of which are Asymptotic Giant Branch (AGB) stars. The variable stars are concentrated towards the centre of M33, more so than low-mass, less-evolved red giants. Our data were matched to optical catalogues of variable stars and carbon stars and to mid-infrared photometry from the Spitzer Space Telescope. Most dusty AGB stars had not been previously identified in optical variability surveys, and our survey is also more complete for these types of stars than the Spitzer survey. The photometric catalogue is made publicly available at the Centre de Donnees astronomiques de Strasbourg.
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