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A recent study associates carbon with single photon emitters (SPEs) in hexagonal boron nitride (h-BN). This observation, together with the high mobility of carbon in h-BN, suggests the existence of SPEs based on carbon clusters. Here, by means of density functional theory calculations, we studied clusters of substitutional carbon atoms up to tetramers in h-BN. Two different conformations of neutral carbon trimers have zero-point line energies and shifts of the phonon sideband compatible with typical photoluminescence spectra. Moreover, some conformations of two small C clusters next to each other result in photoluminescence spectra similar to those found in the experiments. We also showed that vacancies are unable to reproduce the typical features of the phonon sideband observed in most measurements because of the large spectral weight of low-energy breathing modes, ubiquitous in such defects.