Extremal theory for arithmetic forbidden cycle-length sets

Determine natural arithmetic conditions on a set of forbidden cycle lengths under which the extremal number is asymptotically given by the larger of the clique-block and bounded-side complete-bipartite constructions, and characterize the corresponding extremal and near-extremal graphs; in particular, determine when the least odd and least even forbidden lengths alone determine the extremal structure.

Background

The paper proves a sharp Erdős–Gallai-type result for graphs avoiding sufficiently many consecutive even cycle lengths and derives applications to cycle lengths in prescribed residue classes. In the concluding discussion, the authors propose a broader extremal framework for an arbitrary forbidden set of cycle lengths $\mathcal L$. They identify two canonical constructions: connected graphs whose blocks are complete graphs of bounded order, and complete bipartite graphs with one bounded side. These constructions have asymptotic edge densities determined by the least forbidden odd and even lengths, but the paper does not establish conditions ensuring that no other extremal mechanism can dominate. The unresolved problem is therefore to identify arithmetic hypotheses on $\mathcal L$ that force the proposed density formula and the associated clique-block versus complete-bipartite structural dichotomy.