Almost-linear Zarankiewicz bounds in $1$-semi-equational theories

What the math gives to ML

The paper provides incidence bounds that turn a structural restriction on learned relations—being free of a fixed complete bipartite or multipartite pattern—into an almost-linear edge budget. For a binary relation represented by a Boolean combination of m structured predicates, the number of active edges is only linear up to a polylogarithmic factor, rather than quadratic. This suggests sparse attention or retrieval layers whose connectivity is generated by hierarchical, laminar predicates and explicitly prevented from forming large bicliques. The theorem does not guarantee that such sparsification preserves task accuracy, so the right test is whether the resulting mask achieves lower attention FLOPs at matched perplexity or accuracy.