Matching-size-dependent spanning-tree threshold

Determine the infimum of the relative minimum \(\ell\)-degree threshold \(\delta\) such that every sufficiently large \(k\)-uniform hypergraph \(G\) with \(\overline{\delta}_{\ell}(G)\ge\delta+\gamma\) contains every spanning \(k\)-loose tree whose largest matching has size \(\theta n\) and whose maximum vertex degree is at most \(\Delta\), thereby determining whether the threshold depends on the matching number of the tree.

Background

The paper notes that different spanning loose trees impose different matching requirements: binary loose trees of even depth contain perfect matchings, whereas loose Hamiltonian paths need not. Since minimum-degree conditions can force loose Hamiltonian paths below the perfect-matching threshold, the authors ask whether the appropriate embedding threshold is governed by the largest matching in the target tree.

The proposed problem introduces a parameter θ∈[0,1]\theta\in[0,1], representing a largest matching of size θn\theta n, and seeks the corresponding asymptotic degree threshold.

References

Hence, we ask whether the minimum degree threshold for the existence of embedding for a loose tree~$T$ depends on the size of the largest matching in~$T$. \begin{problem} Let $1/n\ll \gamma \ll 1/\Delta, 1/k$, $\ell \in [k-1]$ and $\theta\in [0,1]$. Determine the infimum $\delta$ such that if $G$ is a $k$-graph on the $n$ vertices with~$\overline{\delta}_{\ell}(G)\ge \delta+\gamma$ and $T$ is a $n$-vertex $k$-loose tree with the largest matching size~$\theta n$ and $\Delta_1(T)\le \Delta$, then there exists an embedding from~$T$ to~$G$. \end{problem}

— Embedding loose trees in $k$-uniform hypergraphs  (2502.04783 - Chen et al., 7 Feb 2025) in Section 10.1, “Almost Spanning loose trees in general hypergraphs”