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

Given a graph $F$ and a positive integer $n$, the weak $F$-saturation number $\mathrm{wsat}(K_n,F)$ 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. If $W$ is a vector space spanned by vectors $w(e)$ assigned to edges $e$ of $K_n$ in such a way that, for every copy $F'\subset K_n$ of $F$, there exist non-zero $\lambda_e$, $e\in E(F')$, satisfying $\sum_{e\in E(F')}\lambda_e w(e)=0$, then $\mathrm{dim}W\leq \mathrm{wsat}(K_n,F)$. In this paper, we prove limitations of this approach: we show infinitely many $F$ such that, for every vector space $W$ as above, $\mathrm{dim}W<\mathrm{wsat}(K_n,F)$. We also suggest a modification of this approach that allows to get tight lower bounds even when the original linear algebraic approach is not sufficient. Finally, we generalise our results to random graphs, complete multipartite graphs, and hypergraphs.