Largest equivalent matching extension of a clique

Determine asymptotically the largest integer k=k(s,t), for s<t, such that K_t and K_t+kK_s have the same minimal unbiased Maker–Breaker-winnable graphs.

Background

Ramsey-theoretic results show different behavior for adding disjoint copies of smaller cliques: K_t is Ramsey-equivalent to K_t+K_{t-1} in certain ranges but not to K_t+2K_{t-1}. The authors ask for the analogous asymptotic threshold in the Maker–Breaker setting.

References

Given s,t\in \mathbb{N} with s < t. Determine asymptotically the largest value k=k(s,t) such that K_t K_t + k K_s.

— Ramsey-type results for Maker-Breaker games  (2609.02588 - Allin et al., 2 Sep 2026) in Problem following the discussion of K_t+kK_s, subsection “Equivalent graphs for cliques,” Concluding remarks and open problems