Classify large t-intersecting families below the current threshold

Classify all maximal t-intersecting families in \(\binom{[n]}{k}\) with size at least \(|\mathcal{L}(n,k,t,k-t-1)|\) when \(n=\Omega(k^3)\), thereby extending the classification below the range covered by Theorem 3.1.

Background

The paper establishes a classification of maximal non-trivial $t$-intersecting families whose size is at least $\binom{n-t}{k-t}-\binom{n-k}{k-t}$, under the assumptions $k\geq t+4\geq6$ and $n\geq t+6\max\{(t+2)^2,k(k-t)\}$. The authors identify this as a stability and structural extension of the Hilton–Milner–Frankl theorem.

The proposed problem lowers the size threshold to $|\mathcal{L}(n,k,t,k-t-1)|$ and asks for a complete classification when $n=\Omega(k^3)$. The authors note that extending the classification below the range of their theorem, particularly for smaller ground sets such as $n=O(k^2)$, may be challenging; the stated problem specifies the concrete unresolved regime they propose to study.