Inverse Turán numbers

For given graphs G and F, the Turán number ex(G,F) is defined to be maximum of edges in an F-free subgraph G. Foucaud, Krivelevich Perarnau later independently Briggs Cox introduced a dual version this problem wherein for k, one maximizes host graph which ex(G,H)<k. Addressing Cox, we determine asymptotic value inverse paths length 4 5 provide improved lower bound all even length. Moreover, obtain bounds on cycles giving leading coefficient case C4. Finally, give multiple conjectures concerning C4 P?, suggesting that latter behavior depends heavily parity ?.