General sparse-deletion meta-theorem

Determine whether, for each function $p^*(n)$, there exist a sufficiently broad class of monotone properties and a corresponding notion of $p^*$-sparseness such that a sparse graph can be deleted from a random graph without changing the asymptotic threshold behavior for every property in the class.

Background

The paper develops a property-specific result showing that deleting a sparse graph does not prevent the appearance of the linkage structures needed in its Hamilton-cycle-power construction. It asks for a general framework that would make such arguments reusable across many monotone properties and multi-stage strategies.

References

Given a function $ p* = p*(n) $, does there exist a class $ \mathcal{C} $ of monotone properties, which is large enough to include all natural properties with threshold at most $ p* $ which one might consider in the multi-stage process, as well as a notion of $p*$-sparseness, such that a.a.s\ $ G(n, p* n{o(1)}) $ is $p*$-sparse and, for any property $ \mathcal{P} \in \mathcal{C} $ and any $p*$-sparse graph $ H $, a.a.s\ $ G(n, p) \setminus H $ satisfies $ \mathcal{P} $ whenever $ p = \omega(p*) $?

— Graph factors and powers of Hamilton cycles in the budget-constrained random graph process  (2502.20870 - Díaz et al., 28 Feb 2025) in Problem 7.3, Section 7 “Final remarks and open problems”