Equality of VC-dimension and blowup thresholds

Establish that the VC-dimension threshold and the blowup threshold are equal for every graph H, namely, prove that δ_VC(H)=δ_B(H).

Background

The paper introduces δ_VC(H) as the infimum of the minimum-degree proportions for which every maximal H-free graph has VC-dimension bounded in terms of the density parameter and H. It observes that δ_VC(H)≤δ_B(H), because a bounded-size blowup has bounded VC-dimension.

For odd cycles, the paper proves only the upper bound δVC(C{2k-1})≤1/(2k-1) and explicitly states that equality remains open. It then proposes the broader conjecture that the VC-dimension and blowup thresholds coincide for every graph.

References

It remains open whether δVC(C{2k-1})=1/(2k-1). In general, for any graph H, if a maximal H-free G is a blowup of another graph of bounded size, then G obviously has bounded VC-dimension, therefore δ_VC(H)≤δ_B(H) always holds. We suspect that they always equal.

— Interpolating chromatic and homomorphism thresholds  (2502.09576 - Huang et al., 13 Feb 2025) in Concluding remarks, Section 5.2, “When does a maximal H-free graph have bounded VC-dimension?”, immediately after Problem 5.1