Given a graph F and a positive integer n, the weak F-saturation number \({\textrm{wsat}}(K_n,F)\) 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 \(K_n\) . Suppose that, for every copy \(F'\subset K_n\) of F, there exist non-zero scalars \(\lambda _e\) , \(e\in E(F')\) , satisfying \(\sum _{e\in E(F')}\lambda _e w(e)=0\) . Then \(\textrm{dim}W\le {\textrm{wsat}}(K_n,F)\) . In this paper, we prove limitations of this approach: we find infinitely many F such that, for every vector space W as above, \(\textrm{dim}W<{\textrm{wsat}}(K_n,F)\) . 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.
