We analyse a unique sample of 517 bright ($L>L^{*}$) LBGs at redshift z$sim$3 in order to characterise the distribution of their UV slopes $beta$ and infer their dust extinction under standard assumptions. We exploited multi-band observations over 750 arcmin$^2$ of the COSMOS field that were acquired with three different ground-based facilities: the Large Binocular Camera (LBC) on the Large Binocular Telescope (LBT), the Suprime-Cam on the SUBARU telescope, and the VIRCAM on the VISTA telescope (ULTRAVISTA DR2). Our multi-band photometric catalogue is based on a new method that is designed to maximise the signal-to-noise ratio in the estimate of accurate galaxy colours from images with different point spread functions (PSF). We adopted an improved selection criterion based on deep Y-band data to isolate a sample of galaxies at $zsim 3 $ to minimise selection biases. We measured the UV slopes ($beta$) of the objects in our sample and then recovered the intrinsic probability density function of $beta$ values (PDF($beta$)), taking into account the effect of observational uncertainties through detailed simulations. The galaxies in our sample are characterised by mildly red UV slopes with $<beta>simeq -1.70$ throughout the enitre luminosity range that is probed by our data ($-24lesssim M_{1600}lesssim -21$). The resulting dust-corrected star formation rate density (SFRD) is $log(SFRD)simeq-1.6 M_{odot}/yr/Mpc^{3}$, corresponding to a contribution of about 25% to the total SFRD at z$sim$3 under standard assumptions. Ultra-bright LBGs at $z sim 3$ match the known trends, with UV slopes being redder at decreasing redshifts, and brighter galaxies being more highly dust extinct and more frequently star-forming than fainter galaxies. [abridged]
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