We present a new deep (down to V ~ 24) photometry of a wide region (6x 6) around the LMC globular cluster NGC1866: our sample is complete, down to 3 mag below the brightest MS star. Detailed comparisons with various theoretical scenarios using models computed with the evolutionary code FRANEC have been done reaching the following conclusions: both standard models (i.e. computed by adopting the Schwarzschild criterion to fix the border of the convective core) and models with an enlarged convective core (overshooting) lead to a fair fit of the MS but are not able to reproduce the luminosity and/or the number of He burning giants. Models including a fraction of 30% of binaries leads to a good fit both to the MS luminosity function and to the He clump, if standards models are considered, for a visual distance modulus (m-M)v = 18.8, age t ~ 100 Myr and mass function slope alpha ~ 2.4, thus largely removing the classical discrepancy between observed and predicted number of stars in the He burning clump. The fit obtained with models computed with an enlarged convective core gets worse when a binary component is taken into account, because the presence of binary systems increases the existing discrepancy between the observed and predicted clump luminosity. As a consequence of this analysis, we conclude that the next step towards a proper understanding of NGC 1866, and similar clusters, must include the accurate determination of the frequency of binary systems that will be hopefully performed with the incoming Cycle 8 HST observations of NGC~1866.
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