The asymptotic behaviour of $sat(n,\mathcal{F})$

For a family $\mathcal{F}$ of graphs, $sat(n,\mathcal{F})$ is the minimum number of edges in a graph $G$ on $n$ vertices which does not contain any of the graphs in $\mathcal{F}$ but such that adding any new edge to $G$ creates a graph in $\mathcal{F}$. For singleton families $\mathcal{F}$, Tuza conjectured that $sat(n,\mathcal{F})\mathrm{/}n$ converges and Truszczynski and Tuza discovered that either $sat(n,\mathcal{F})=(1-1\mathrm{/}r)n+o(n)$ for some integer $r\ge 1$ or $sat(n,\mathcal{F})\ge n+o(n)$. This is often cited in the literature as the main progress towards proving Tuza’s Conjecture. Unfortunately, the proof is flawed. We give a correct proof, which requires a novel construction. Moreover, for finite families $\mathcal{F}$, we completely determine the possible asymptotic behaviours of $sat(n,\mathcal{F})$ in the sparse regime $sat(n,\mathcal{F})\le n+o(n)$. Finally, we essentially determine which sequences of integers are of the form ${(sat(n,\mathcal{F}))}_{n\ge 0}$ for some (possibly infinite) family $\mathcal{F}$.