Asymptotic Erdős–Gallai theorem for hypergraph matchings

Determine whether the inequality in the trivial lower bound for the extremal number of tilings by the complete $r$-uniform hypergraph $K_{1,\\ldots,1}^{r}$ holds with equality for all $r\ge 3$, thereby extending the Erdős–Gallai Theorem from graphs to higher-uniformity hypergraphs.

Background

The paper studies extremal numbers of the form $\mathrm{ex}(n,(k+1)F)$, where the host $r$-graph is required to have matching number below $k+1$ with respect to a fixed $r$-graph $F$. For complete $r$-partite $r$-graphs, the authors introduce a family of natural constructions $G_{n,i,\beta}(s_1,\ldots,s_r)$ that provide lower bounds for these extremal numbers.

When $F=K_{1,\ldots,1}^{r}$, the relevant construction has two competing extremal forms, corresponding to $i=1$ and $i=r$. The Erdős–Gallai Theorem proves equality in the analogous lower bound for ordinary graphs ($r=2$), while the higher-uniformity extension is identified as a major unresolved problem.