Determine the threshold for bounds on the (ell+1)th largest degree

Determine or estimate the minimum value n_0(k,ell) such that, for every n>n_0(k,ell), the (ell+1)th largest degree d_{ell+1}(F) of every intersecting k-uniform hypergraph F on n vertices satisfies d_{ell+1}(F)\leq\binom{n-2}{k-2}+\binom{n-ell-1}{k-ell}, for 4\leqell\leq k+1.

Background

The paper studies the ordered vertex degrees d_1(F)\geq\cdots\geq d_n(F) of intersecting k-uniform hypergraphs. It establishes sharp or near-sharp upper bounds for several positions in this ordering, including d_2, d_3, d_4, d_{\lceil 8k/3\rceil}, d_{2k+1}, and d_{k+2} under various lower bounds on n.

For the general range 4\leq\ell\leq k+1, the authors identify the benchmark bound \binom{n-2}{k-2}+\binom{n-\ell-1}{k-\ell}, which is attained in the relevant degree positions by the natural families H_\ell. The stated problem asks for the smallest asymptotic or exact threshold n_0(k,\ell) beyond which this inequality must hold for every intersecting k-graph.