Tuza's conjecture on the bounded lower-order term

Establish that for every finite graph F, there exists a constant C_F such that, for all sufficiently large n, the weak saturation number satisfies |wsat(n,F)-w_F n|≤C_F, equivalently w_F n-C_F≤wsat(n,F)≤w_F n+C_F.

Background

For every graph F, the weak saturation number has an asymptotic form wsat(n,F)=w_F n+o(n), where w_F is the weak saturation limit. Tuza conjectured that the error term can always be bounded independently of n. The paper notes that the lower inequality is straightforward and that its proof establishes the conjecture in the sparse regime w_F<δ(F)/2, but the conjecture remains unresolved in general.