Almost-spanning loose-tree threshold in general hypergraphs

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 \((1-\varepsilon)n\)-vertex \(k\)-loose tree of maximum vertex degree at most \(\Delta\), and determine in particular whether \(\delta<\delta^{T}_{k,k-1}=1/2\) when \(\ell=k-1\).

Background

The paper’s spanning result uses an absorption step, which imposes the threshold $1/2$. For almost-spanning trees, the authors observe that a lower threshold may suffice because the unused vertices could serve as a reservoir without requiring a spanning absorber.

They formulate a threshold problem for embedding bounded-degree loose trees on (1−ε)n(1-\varepsilon)n vertices in general kk-uniform hypergraphs and specifically ask whether the codegree threshold can be strictly below the spanning threshold.

References

So we wonder whether the relative minimum degree threshold for existence of almost spanning loose trees is smaller than the threshold for spanning loose trees. \begin{problem} Let $1/n\ll \varepsilon \ll \gamma \ll 1/\Delta, 1/k$ and $\ell \in [k-1]$. Determine infimum $\delta$ such that if $G$ is a $k$-graph on~$n$ vertices with~$\overline{\delta}{\ell}(G)\ge \delta+\gamma$ and~$T$ be a $k$-loose tree on~$(1-\varepsilon)n$ vertices with~$\Delta_1(T)\le \Delta$, then there is an embedding from~$T$ to~$G$. In particular, when $\ell = k-1$, is $\delta<\delta{T}{k,k-1}=1/2$? \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”