Pehova–Petrova threshold conjecture

Establish that the loose-tree embeddable threshold equals the perfect-matching threshold, namely, prove that \(\delta_{k,\ell}^{T}=\delta_{k,\ell}^{PM}\) for every pair of integers \(1\le \ell<k\).

Background

The paper defines $\delta_{k,\ell}^{T}$ as the asymptotic relative minimum $\ell$-degree threshold forcing a $k$-uniform hypergraph to contain every spanning $k$-loose tree of bounded maximum vertex degree. It defines $\delta_{k,\ell}^{PM}$ as the corresponding threshold forcing a perfect matching.

Pehova and Petrova conjectured that these thresholds coincide for all $1\le \ell<k$. The paper verifies the conjecture for $(k,\ell)=(k,k-1)$, $(3,1)$, and, through its main theorem, for all $\ell=k-2$ with $k\ge4$; the general case remains unresolved.