Resolve the boundary case of the dense 3-uniform hypergraphicality dichotomy

Determine the computational complexity and realizability characterization for degree sequences satisfying the critical density condition c_1=c_1^*(c_2) in the dense 3-uniform hypergraphical degree sequence problem.

Background

The paper summarizes an earlier dichotomy theorem for dense 3-uniform hypergraphical degree sequences. For fixed c_2, sequences whose degrees lie between c_1 and c_2 times the maximum possible vertex degree are always realizable, subject to divisibility, when c_1>c_1*(c_2), while deciding realizability is NP-complete when c_1<c_1*(c_2). The equality case at the threshold is explicitly left unresolved, so the earlier result does not constitute a strict dichotomy at that boundary.

References

Note that this is not a strict dicghotomy theorem for the case $c_1 = c_1*(c_2)$ is not resolved.

— A complete dichotomy theorem on the sparse $t$-Uniform Hypergraphicality Problem  (2512.15356 - Miklós et al., 17 Dec 2025) in Section 1, Introduction