2026/07/23 by Fan Chang, Hong Liu, Miao Liu
#math.CO #math.CA #math.FA
We settle the uniform and biased product conjectures of Frankl and Tokushige for r-cross-intersecting families. Let r≥2, let 0≤ ki≤(r-1)n/r, and let Fi⊆\binom[n]ki be r-cross-intersecting. We prove the sharp inequality ∏i=1r\frac|Fi|\binomnki≤ ∏i=1r(ki)/(n), with equality attained by the corresponding levels of a common 1-star. As a consequence, we obtain the analogous pi-biased measure theorem for 0≤ pi≤(r-1)/r, ∏i=1rμpi(Fi)≤ ∏i=1r pi.The main difficulty is that unequal parameters do not determine a single common target level; instead, the target levels ℓ1,…,ℓr must satisfy ∑i=1r ℓi=(r-1)n. We overcome this asymmetry in three steps. An ordered-partition coupling gives a sharp additive inequality for every such choice of target levels. A star-calibrated upper-shadow inequality relates the density of a family on its original level to the density of its upper shadow on a suitably chosen target level; it is proved by induction on n, with the induction step reduced to a two-point inequality. Finally, an analytic inequality shows that the resulting asymmetric additive estimate implies the required product bound. Perhaps surprisingly, the coupling captures all the combinatorial information of cross-intersection, reducing the remainder of the proof to an analytic argument.