Stability and structure of near-extremal graphs for ascending-path containment

Characterize the structure of all graphs \(G\), for fixed \(k>2\) and \(\varepsilon>0\), such that every subgraph \(G'\) with \(e(G')>(\frac{k-1}{2k}+\varepsilon)e(G)\) contains a copy of the ascending path \(P_k\), and determine whether such graphs must resemble the recursively constructed graphs \(G_\varepsilon(n,d)\); additionally, determine whether a density assumption \(e(G)\ge \varepsilon v(G)^2\) should be imposed.

Background

Theorem 1.2 determines the relative Turán density of the ascending path PkP_k. Problem 4.3 asks for a stability theorem describing host graphs in which every sufficiently dense subgraph necessarily contains PkP_k. The paper specifically raises whether the recursively defined graphs Gε(n,d)G_\varepsilon(n,d), used to prove the upper bound in Theorem 1.2, provide the model for all such near-extremal host graphs, and whether one should restrict attention to dense graphs.

References

Problem 4.3. Given k > 2 and = > 0, describe the structure of all graphs G with the property that every subgraph G' satisfying e(G') > (5-1 + €)e(G) contains a copy of Pk. In particular, one may ask whether such graphs need to have any resemblance to Gn (n, d) for some small n = 1(k,¿). Perhaps one should also assume here that G be dense, i.e., that e(G) ≥ Ev(G)2.

— Relative Turán densities of ordered graphs  (2501.06853 - Reiher et al., 12 Jan 2025) in Problem 4.3, Section 4 (Concluding Remarks)