Suppose $C$ is weakly sparse and monadically dependent but not nowhere dense. Passing to the closure of the class under graph copies and to the shallow-minor formulation, some radius $r$ admits $r$-subdivided cliques of every order as subgraphs of members. At $r = 0$ these are cliques, which contain large bicliques, contradicting weak sparseness. At $r ≥ 1$ a subdivided clique contains a subdivided biclique of half the order, and the Ramsey-theoretic extraction of Lemma 13.8 of Mählmann's thesis turns those into either large bicliques — again contradicting weak sparseness — or induced $r'$-subdivided bicliques with $1 ≤ r' ≤ r$. A pigeonhole fixes one $r'$ occurring at unbounded orders, and the star-crossing transduction then transduces all graphs from $C$, contradicting monadic dependence.

The two Ramsey inputs are assumed from the Finite Ramsey submission rather than reproved: the monochromatic-subset extraction inside the nowhere-dense bridge is the multicolour Ramsey statement, and the order-type homogeneity behind Lemma 13.8 is the tuple Ramsey statement.

Corollary 6a of Dreier, Mählmann, McCarty, Pilipczuk and Toruńczyk, Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes (2026). The argument is Lemma 13.7 of Mählmann's thesis, through its Lemma 13.8, with the forbidden-pattern endpoint replaced by a transduction of all graphs.