Size-Maker–Breaker number of paths

Determine the size-Maker–Breaker number \hat r_{MB}(P_n)=\min\{e(G):G\,MB_1\,P_n\} for every n.

Background

The paper introduces \hat r_{MB}(H) as the minimum number of edges in a graph on which Maker wins the unbiased H-game. A construction gives \hat r_{MB}(P_n)\leq4n-11 for n\geq4, but the authors explicitly do not know whether this bound is asymptotically optimal.

References

The simple construction in Figure~\ref{fig:size.path.construction} can be used to show that \hat{r}_{MB}(P_n) \leq 4n-11$ for $n\geq 4$, but we do not know whether this is asymptotically best possible.

— Ramsey-type results for Maker-Breaker games  (2609.02588 - Allin et al., 2 Sep 2026) in Problem following the discussion of \hat r_{MB}(P_n), subsection “A game analogue of size-Ramsey numbers,” Concluding remarks and open problems