On the product of nontrivial cross-intersecting families

Let n≥2≥2k≥6 and n>k+. We determine the maximum of | A|| B| over all nonempty, nontrivial cross-intersecting families A⊆[n]k and B⊆[n]. The exact bound is | A|| B|≤max₁,M₂\, where \[ {aligned} M_1 &=({n-1}{k-1}-{n--1}{k-1}+1) ({n-1}{-1}-{n-k-1}{-1}+1),\\ M_2 &=({n-2}{-2}+1) ∑ᵢ₌₁² ({n-i}{k-1}-{n--i}{k-1}). {aligned} \] Both terms are attained by explicit constructions: a Hilton--Milner-type pair and a two-point construction, respectively. We classify all equality cases, including the additional cross-triangle pairs when k==3. Two structural results underlie the proof. First, we establish a restricted-lex comparison theorem showing that restricted lexicographic initial segments minimize disjointness neighborhoods, and characterize equality at the first nontrivial threshold. Second, a decomposition of critically shifted families into two parts with disjoint cores leads to a sharp product theorem, with equality characterized, for all cross-intersecting pairs admitting this decomposition, without additional nontriviality assumptions. Endpoint comparisons and reverse shifting complete the proof for unequal as well as equal uniformities.