Weak saturation rank: a failure of the linear algebraic approach to weak saturation

Given a graph F and a positive integer n, the weak F-saturation number wsat(Kn, F) is the minimum number of edges in a graph H on n vertices such that the edges missing in H can be added, one at a time, so that every edge creates a copy of F. Kalai in 1985 introduced a linear algebraic approach that became one of the most efficient tools to prove lower bounds on weak saturation numbers. Let W be a vector space spanned by vectors w(e) assigned to edges e of Kn. Suppose that, for every copy F ⊂ Kn of F, there exist non-zero scalars λe, e ∈ E(F), satisfying e∈E(F) λew(e) = 0. Then dimW ≤ wsat(Kn, F). In this paper, we prove limitations of this approach: we find infinitely many F such that, for every vector space W as above, dimW < wsat(Kn, F). We also introduce a modification of this approach that yields tight lower bounds even when the original direct approach is insufficient. Finally, we generalise our results to random graphs, complete multipartite graphs, and hypergraphs.