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, A near-exponential improvement of a bound of Erd˝ os and Lov´ asz on max- imal intersecting families, Combinatorics, Probability & Computing 28(5) (2019), 733–739
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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.