Determine the optimal reconstruction density for F-tiling polynomials

Determine, for each fixed graph F, the minimum constant 0 ≤ c_F < 1 such that for every ε > 0 there exists n_0 = n_0(F, ε) for which the F-tiling polynomial of every n-vertex graph G with n ≥ n_0 is uniquely determined by the multiset of induced subgraphs of G on ⌊(c_F + ε)n⌋ vertices.

Background

The paper proves that if F is a k-vertex graph and G is an n-vertex graph, then the F-tiling polynomial of G is determined by the multiset of induced subgraphs on ⌊(k−1)n/k⌋ + 1 vertices. This yields the upper bound c_F ≤ (|V(F)|−1)/|V(F)| for the asymptotic reconstruction threshold defined in the problem.

The authors explicitly state that it is unknown whether this bound is essentially tight. For F = K_2, the threshold is known to be c_F = 1/2, but the optimal value remains unresolved for general graphs F.