  Understanding the formation mechanisms of protoplanetary disks and multiple
systems, and their pristine properties, is a key question for modern
astrophysics. The properties of the youngest disks, embedded in rotating
infalling protostellar envelopes, have largely remained unconstrained up to
now.
  In the framework of the IRAM-PdBI CALYPSO survey, we have obtained
sub-arcsecond observations of the dust continuum emission at 231 GHz and 94
GHz, for a sample of 16 solar-type Class 0 protostars. In an attempt to
identify disk-like structures embedded at small scales in the protostellar
envelopes, we model the dust continuum emission visibility profiles using both
Plummer-like envelope models and envelope models including additional Gaussian
disk-like components.
  Our analysis shows that in the CALYPSO sample, 11 of the 16 Class 0
protostars are better reproduced by models including a disk-like dust continuum
component contributing to the flux at small scales, but less than 25% of these
candidate protostellar disks are resolved at radii > 60 au. Including all
available literature constraints on Class 0 disks at subarcsecond scales, we
show that our results are representative: most (> 72% in a sample of 26
protostars) Class 0 protostellar disks are small and emerge only at radii < 60
au. Our multiplicity fraction at scales 100-5000 au is in global agreement with
the multiplicity properties of Class I protostars at similar scales.
  We confront our observational constraints on the disk size distribution in
Class 0 protostars to the typical disk properties from protostellar formation
models. Because they reduce the centrifugal radius, and produce a disk size
distribution peaking at radii <100 au during the main accretion phase,
magnetized models of rotating protostellar collapse are favored by our
observations.
