Limit of normalized counting thresholds

Establish whether, for every pair of integers 1≤d≤k−1, the ratio c_d(k,n)/\binom{n-d}{k-d} converges as n tends to infinity.

Background

The d-degree counting threshold c_d(k,n) is the least minimum d-degree forcing the entropy lower bound corresponding to the expected number of perfect matchings in a random hypergraph with the same edge density. Unlike the known convergence result for Dirac thresholds, the paper obtains only liminf- and limsup-based information for these counting thresholds and therefore introduces approximate counting thresholds β_d(k). Determining convergence would clarify whether the exact counting thresholds possess a well-defined asymptotic normalized value.

References

One can ask whether this is sharp, but unfortunately, we believe the answer is no.

— Counting thresholds for perfect matchings in hypergraphs  (2608.19345 - Gvozdić, 19 Aug 2026) in Section Concluding remarks, Section 4

Unlike the case of Dirac thresholds, here we are not able to show that the ratio $c_d(k,n)/\binom{n-d}{k-d}$ tends to a limit as $n$ goes to infinity.

— Counting thresholds for perfect matchings in hypergraphs  (2608.19345 - Gvozdić, 19 Aug 2026) in Section 1, Introduction; reiterated at the end of the proof of the theorem bounding beta_d(k)