The tripartite hypergraph degree-sequence threshold conjecture

Let $c$ be the positive real number specified by

$$c = 8 - \frac{3}{2} (1 + i \sqrt{3})\sqrt[3]{\frac{1}{2} (37 + i \sqrt{3})} - \frac{21 (1 - i \sqrt{3})}{\sqrt[3]{4 (37 + i \sqrt{3})}}\approx 0.278066.$$

A tripartite hypergraph degree sequence on $n+n+n$ vertices consists of degree data for the three vertex classes, and it is graphic if it has a tripartite hypergraph realization. The tripartite hypergraph degree-sequence threshold conjecture. For every $\varepsilon>0$, sufficiently large $n$, every tripartite hypergraph degree sequence on $n+n+n$ vertices with equal degree sums in the three vertex classes and all degrees between $(c+\varepsilon)n^2$ and $(1-c-\varepsilon)n^2$ is graphic. Conversely, for every $\varepsilon>0$ and sufficiently large $n$, there exists such a degree sequence with equal degree sums and all degrees between $(c-\varepsilon)n^2$ and $(1-c+\varepsilon)n^2$ that is not graphic. This conjecture identifies the sharp asymptotic degree range in which balanced tripartite $3$-uniform hypergraph degree sequences are always realizable; the surrounding discussion compares the conjectured threshold with the proved bound $2/7$ and notes technical and rounding difficulties near the limit.