Proper compression for balls in K_t-minor-free graphs

Determine whether there exists a function f(t) such that, for every positive integer t and every K_t-minor-free graph G, the hypergraph of balls in G admits a proper sample compression scheme of size f(t).

Background

While O(t2 log t) (improper) bounds follow from general theory, this paper achieves O(t log t) proper schemes for treewidth-t graphs. Extending proper schemes with controlled size to all K_t-minor-free graphs is a key challenge bridging sparse graph structure and sample compression.

References

Is there a function $f\colon \mathbb{N}\to\mathbb{N}$ such that for every positive integer $t$ and every $K_t$-minor-free graph $G$, the hypergraph of balls in $G$ admits a proper sample compression scheme of size $f(t)$?

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