An ordering of the vertices is a permutation π of Fin m assigning each vertex its position. Weak reachability is stated with walks, as in the nowhere dense concept: shortcutting a walk to a path only shrinks its support, so walks of length at most r with π-minimal endpoint reach exactly the vertices that such paths do. wcol is the least achievable bound k, a Nat.sInf (attained: k = m works for any ordering); the count is Set.ncard as in the neighborhood complexity concept.

The bound is uniform over subgraph copies (⊑, as in the weakly sparse concept) of members, each measured by its own vertex count m. Since nowhere denseness survives taking subgraphs, this is the literature statement for subgraph-closed classes, and the uniformity is what localization arguments downstream consume. At m = 0 both sides vanish, so no nonemptiness hypothesis is needed. The theorem combines Zhu's bounds relating weak coloring numbers to densities of shallow minors with the subpolynomial density bounds for nowhere dense classes (Nešetřil, Ossona de Mendez); see chapters 2 and 5 of the sparsity lecture notes of Pilipczuk, Pilipczuk, Siebertz.