Extend the t-intersection Han–Kohayakawa bound to all parameter ranges

Determine whether there exists an absolute constant \(C\) such that, for every \(k\geq t+2\geq4\) and \(n\geq Ckt\), every non-trivial \(t\)-intersecting family in \(\binom{[n]}{k}\) that is isomorphic to neither a subfamily of \(\mathcal{A}(n,k,t)\) nor a subfamily of \(\mathcal{H}(n,k,t)\) has size at most \(|\mathcal{K}(n,k,t,2)|\).

Background

Corollary 4.2 proves the asserted upper bound ∣F∣≤∣K(n,k,t,2)∣|\mathcal{F}|\leq|\mathcal{K}(n,k,t,2)| under the stronger condition k≥t+4≥6k\geq t+4\geq6 and n≥t+6max⁡{(t+2)2,k(k−t)}n\geq t+6\max\{(t+2)^2,k(k-t)\}. This gives a t-intersection analogue of the Han–Kohayakawa theorem for a substantial large-ground-set range.

The authors observe that their lower bound on nn is of order O((t+1)(k−t+1))O((t+1)(k-t+1)) when k−t=Θ(t)k-t=\Theta(t), and explicitly ask whether a bound of the form n≥Cktn\geq Ckt suffices uniformly for all k≥t+2≥4k\geq t+2\geq4. A positive answer would extend the result to the remaining parameter ranges, including the cases excluded by the stated theorem.

References

By Theorem \ref{thmlarge} and Lemma \ref{lemmacountingfamily}, we obtain Corollary \ref{coroH-K}, which provides a $t$-intersection version of Han--Kohayakawa theorem for large $n$. In particular, the lower bound on $n$ here equals $O((t+1)(k-t+1))$ for $k-t=\Theta(t)$. It would be interesting to consider whether it holds for all $k\geq t+2$. \begin{problem} Is there an absolute constant $C$ such that the following holds for all $k\geq t+2\geq4$ and $n\geq Ckt$? If $\mathcal{F}$ is a non-trivial $t$-intersecting family which is isomorphic to neither a subfamily of $\mathcal{A}(n,k,t)$ nor a subfamily of $\mathcal{H}(n,k,t)$, then $|\mathcal{F}|\leq|\mathcal{K}(n,k,t,2)|$. \end{problem}

— Structure of large $t$-intersecting families I: Stability for the Hilton--Milner--Frankl theorem  (2608.14197 - Wen et al., 14 Aug 2026) in Section 6, Concluding Remarks, immediately after Corollary 4.2