Sharpness of the asymptotic order bound for non-trivial lambda-intersecting hypergraphs

Determine whether, for every fixed lambda and sufficiently large k, every non-trivial lambda-intersecting k-uniform hypergraph has order at most (1+o_k(1))4k^3/(27lambda).

Background

The paper constructs non-trivial lambda-intersecting hypergraphs from symmetric 2-(v,q+lambda,lambda)-designs whose order is asymptotically (1+o(1))4k3/(27lambda). This motivates asking whether the construction gives the correct asymptotic maximum order.

The question is posed for fixed lambda and sufficiently large k and concerns whether the displayed asymptotic expression is a universal upper bound for all non-trivial lambda-intersecting k-uniform hypergraphs.

References

For fixed $lambda$ and sufficiently large $k$, is it true that a non-trivial $lambda$-intersecting $k$-graph has order bounded by $(1+o_k(1))\frac{4k3}{27\lambda}$?

— On the order of intersecting hypergraphs  (2504.05162 - Cambie et al., 7 Apr 2025) in Question, concluding remarks, Section 5

For fixed $\lambda$ and sufficiently large $k$, is it true that a non-trivial $\lambda$-intersecting $k$-graph has order bounded by $(1+o_k(1))\frac{4k3}{27\lambda}$?

— On the order of intersecting hypergraphs  (2504.05162 - Cambie et al., 7 Apr 2025) in Concluding remarks, Question