Any intersecting k-graph with covering number at least three satisfies the stated binomial-coefficient upper bound for all n greater than or equal to 2k.
Frankl, On the maximum of the sum of the sizes of non-trivial cross-intersecting families, Combinatorica 44 (2024), 15–35
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Intersecting families with covering number three II
Any intersecting k-graph with covering number at least three satisfies the stated binomial-coefficient upper bound for all n greater than or equal to 2k.