2026/06/30 by Jianfeng Hou, Caiyun Hu
#math.CO
We study non-trivial intersection problems for multi-partite hypergraphs, excluding the usual extremal examples determined by fixed vertices or fixed coordinates. Our first result determines the exact value of the non-trivial t-intersection problem in the symmetric product [n]r for 1≤ t≤ r-2 and all n≥2. Frankl and Nie proved a two-candidate formula for sufficiently large n and conjectured it for all n≥ 2; our formula shows that the conjectured expression must be enlarged, in small ranges of n, by additional ball-type terms arising from the Frankl families. Our second result concerns intersecting families in general products X1×⋯× Xr, where |Xi|=ni, with no common vertex. Let m0(1,n1,…,nr) denote the largest size of such a family. We show that this number is equal to the maximum of ∑X∈ D∏i∈ X(ni-1) over all downsets D⊆ 2[r] such that \bigcupX∈ DX=[r] and no two members of D have union [r]. This finite reduction separates the intersection obstruction from the part sizes and yields explicit fully asymmetric formulas for r=4,5,6.