General multi-stage coupling theorem

Develop a version of the two-stage coupling lemma that applies to arbitrary multi-stage strategies with a constant number of stages.

Background

The paper introduces a coupling result for analyzing multi-stage strategies, but explains that the result is limited to settings in which only two stages interact through non-disjoint edge sets. A broader theorem would support analyses of more general adaptive strategies.

References

Is it possible to develop a version of \cref{lem:two_stage_real} which holds for arbitrary multi-stage strategies with a constant number of stages?

— Graph factors and powers of Hamilton cycles in the budget-constrained random graph process  (2502.20870 - Díaz et al., 28 Feb 2025) in Problem 7.2, Section 7 “Final remarks and open problems”