Determine asymptotic connected bipartite Turán numbers for paths

Determine, at least asymptotically, \(\operatorname{ex}_{b,c}(n,P_k)\) for every fixed path order \(k\).

Background

The paper proves the exact identity ex⁡b,c(n,P5)=ex⁡b,c(n,P6)=2n−1\operatorname{ex}_{b,c}(n,P_5)=\operatorname{ex}_{b,c}(n,P_6)=2n-1. It leaves open the asymptotic behavior for connected bipartite graphs avoiding an arbitrary fixed path PkP_k, presenting this as a direct extension of the established cases.

References

Problem 16. Determine at least asymptotically exb,c(n, Pk) for any fixed k. This would be an extension of Theorem 7 (1) that establishes exb,c(n, P5) = exb,c(n, P6) = 2n - 1.

— Bipartite Turán number of trees  (2502.09052 - Caro et al., 13 Feb 2025) in Problem 16, Section 4, “Concluding Remarks and Open Problems”