Locate the transition between the two Erdős Matching Conjecture extremal examples

Determine, for each relevant pair of parameters \(k\) and \(s\), the exact value or criterion of \(n\) at which the transition occurs between the two extremal constructions in the Erdős Matching Conjecture.

Background

The Erdős Matching Conjecture proposes two competing extremal constructions for a k-uniform family with matching number at most s: a family supported on a bounded initial segment and a family consisting of all k-sets meeting a fixed s-element set. The dominant construction changes as n varies.

The paper notes that the precise transition point between these constructions is not known, even in the classical matching case. Determining this threshold would sharpen the understanding of when each candidate is extremal.