For k-subset families that cross-intersect but each lacks a common element, their size product is bounded by the square of the Hilton-Milner number when k is at least 8.
Frankl, The maximum of the product of non-trivial cross-intersecting 3-graphs, Combinatorics and Number Theory 15(1)(2026) 1–8
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A product version of the Hilton-Milner Theorem II
For k-subset families that cross-intersect but each lacks a common element, their size product is bounded by the square of the Hilton-Milner number when k is at least 8.