Combinatorial analysis proves an upper bound on product sizes in non-trivial cross-intersecting set families, confirming the Hilton-Milner product conjecture across the full parameter range.
Two families F,G of k-subsets of \1,2,…,n\ are called { non-trivial cross-intersecting} if F∩ G≠ ∅ for all F∈ F, G∈ G and ∩ F∈ F\=∅=∩ G\. In this note, we establish the product version of the Hilton-Milner Theorem for k≥ 8 in the full range. That is, if F,G⊂ [n]k are non-trivial cross-intersecting, n≥ 2k+1 and k≥ 8, then \[ |F||G|≤ ({n-1}{k-1}- {n-k-1}{k-1} +1)^2. \]
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Frankl et al. (2026) studied this question.
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