An incidence approach to cross-intersection in semilattices

Source abstract

We present an incidence-based method for product bounds on cross-intersecting families in finite graded meet-semilattices. The two families may lie in different rank or type fibres and may have different design strengths. The criterion separates global incidence parameters from two directional local counts and yields the sharp product bound, with equality only for full stars with a common centre. Applications include mixed subspace types relative to a fixed subspace, a type-restricted generator family in $Q^-(7,q)$, and mixed form ranks in symplectic spaces. The same short criterion gives several sharp results not obtained directly from the specific theorems and general criteria compared here. In the singular-space case, it also applies in an infinite parameter range beyond the natural star-ratio and weak-spreadness tests.