Interval conjectures for level Hilbert functions

We conjecture that the set of all Hilbert functions of (artinian) level algebras enjoys a very natural form of regularity, which we call the Interval Conjecture (IC): If, for some positive integer α, $(1,h_1,\dots ,h_i,\dots ,h_e)$ and $(1,h_1,\dots ,h_i+\alpha ,\dots ,h_e)$ are both level h-vectors, then $(1,h_1,\dots ,h_i+\beta ,\dots ,h_e)$ is also level for each integer $\beta =0,1,\dots ,\alpha$. In the Gorenstein case, i.e. when $h_e=1$, we also supply the Gorenstein Interval Conjecture (GIC), which naturally generalizes the IC, and basically states that the same property simultaneously holds for any two symmetric entries, say $h_i$ and $h_{e?i}$, of a Gorenstein h-vector.These conjectures are inspired by the research performed in this area over the last few years. A series of recent results seems to indicate that it will be nearly impossible to characterize explicitly the sets of all Gorenstein or of all level Hilbert functions. Therefore, our conjectures would at least provide the existence of a very strong — and natural — form of order in the structure of such important and complicated sets.We are still far from proving the conjectures at this point. However, we will already solve a few interesting cases, especially when it comes to the IC, in this paper. Among them, that of Gorenstein h-vectors of socle degree 4, that of level h-vectors of socle degree 2, and that of non-unimodal level h-vectors of socle degree 3 and any given codimension.