  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. The
most extensive dataset was obtained in the K-band with the UIST instrument for
the central 4'x 4' (1 square kpc) - this contains the nuclear star cluster and
inner disc. These data, taken during the period 2003-2007, were complemented by
J- and H-band images. Photometry was obtained for 18,398 stars in this region;
of these, 812 stars were found to be variable, most of which are Asymptotic
Giant Branch (AGB) stars. Our data were matched to optical catalogues of
variable stars and carbon stars, and to mid-infrared photometry from the
Spitzer Space Telescope. In this first of a series of papers, we present the
methodology of the variability survey and the photometric catalogue - which is
made publicly available at the Centre de Donnees astronomiques de Strasbourg
(CDS) - and discuss the properties of the variable stars. 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.
