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tokens: ones are for the same voids after cleaning. The vertical dotted lines mark the void radius in both cases. We also show the background density level (solid horizontal black line) and the underdensity threshold of equation ( 12 ) computed for the different redshifts (dashed horizontal black line) as a reference. It can be noticed that, while vide void limits reside close to the compensation wall and embed a density that approaches the background, our voids are, by construction, defined in much deeper regions of the density field. These observations have to be kept in mind to understand how the algorithm we apply to clean the cosmic void catalogues affects the void structure. Thanks to this modification, we can be confident that the voids we are using to measure the size function are as close as possible to the objects for which we are able to construct the theoretical model. Fig. 2 compares the measured and modelled void size functions, defined as the comoving number density of voids in logarithmic bins of effective radii, divided by the logarithmic extent of the bins. The four panels show the results at different redshifts. The different symbols in the upper part of each panel show the void size distributions measured in the simulation snapshots at different resolutions, while the gray shaded regions represent the model predictions. The model depends on two thresholds, the one of the shell-crossing, which is fixed at the value , corresponding to our choice of threshold in the void catalogue cleaning, and the overdensity one, δ c , used to account for the void-in-cloud effect. There is a persisting uncertainty on the definition of the latter, since it is not clear if this effect becomes relevant at turn-around (corresponding to a density contrast of δ c ≈ 1.06) or when the overdensity collapses (at a density contrast δ c ≈ 1.69). Since in realistic occurrences we do not expect the spatial resolution of tracers to be high enough to inspect the radii range in which the void size function becomes sensitive to the void-in-cloud effect, we leave the overdensity threshold free to vary in the range 1.06 ≤ δ c ≤ 1.69, thus the shaded region of Fig. 2 . Figure 2. Upper part of each panel: Void size function prediction (gray shaded region) at four different redshifts (different panels) compared to the distribution of cosmic voids after applying the cleaning method described in Section 2.2 (different markers correspond to different simulations, the values of L box in legend corresponding to those of Table 2 ). Lower part of each panel: Logarithmic differences between the measured distribution and the Vdn model prediction. For the largest simulation () we only have snapshots at z = 0 and z = 1, thus the corresponding abundances are missing from the z = 0.5 and z = 1.5 panels. We cut the distribution measured in each of the simulations at twice the m.i.s. of the given simulation box (see Table 2 ). To highlight the difference between the theoretical model