A Sharp Product Bound for Disjoint Cross-Intersecting 3-Graphs with Covering Number Three

Abstract

Lin, Frankl, and Wu proved that the product of the sizes of two cross-intersecting 3-uniform hypergraphs with covering number three is at most 121. They conjectured that requiring the families to be disjoint lowers the sharp bound to 100. We prove this conjecture. The smaller family has at most eleven edges; after fixing it, the other family may be enlarged to its external family of 3-transversals. Splitting by matching number leaves an elementary intersecting case and two finite kernels. We prove the correctness of both encodings and every pruning rule, and exhaustive C++20 computations give the external-blocker maxima 21, 19, 16, 14, 12, 11, 10, 9 for family sizes 4 through 11. Concrete witnesses are checked independently in Python. A pair of ten-edge families on six vertices attains product 100.