Near-linear compression for balls in K_t-minor-free graphs

Determine whether every K_t-minor-free graph G has the hypergraph of balls admitting a sample compression scheme of size \~O(t) (up to polylogarithmic factors).

Background

Applying existing VC-dimension bounds and the general Moran–Yehudayoff construction yields O(t2 log t) for K_t-minor-free graphs, while this paper improves to almost linear size for treewidth-t graphs. Extending an almost-linear bound from treewidth to full minor-closed classes is a natural next step.

References

Is it true that for every $K_t$-minor-free graph $G$, the hypergraph of balls in $G$ admits a sample compression scheme of size $\widetilde{O}(t)$?

— Sample compression schemes for balls in structurally sparse graphs  (2604.02949 - Bourneuf et al., 3 Apr 2026) in Section 7 (Open problems)