Bounded-error linearity of planar Turán numbers

Determine whether, for every planar graph G, there exists a positive integer constant C=C(G) such that the inequality |exP(n,G)−Π(G)n|≤C holds for all positive integers n.

Background

The paper observes that the planar Turán number exP(n,G) is asymptotically governed by the limiting density Π(G). It then asks whether the deviation from the linear asymptotic term can always be bounded by a constant independent of n.

The displayed formulation in the paper uses Π(2C_k) in the inequality and writes C=C(H), although the problem is introduced for an arbitrary planar graph G; the intended issue is whether planar Turán numbers universally admit a linear expression with uniformly bounded additive error. The paper notes that a positive answer would make exP(n,G) approximately linear in n.