This submission states Theorem 2 of Dreier, Mählmann, McCarty, Pilipczuk, Toruńczyk, Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes (2026): every monadically dependent class of finite graphs has almost linear neighborhood complexity — for every $\varepsilon > 0$ there is a $c$ such that every member $G$ and every nonempty vertex subset $A$ satisfy $|\{N(v) \cap A : v \in V(G)\}| \le c\,|A|^{1+\varepsilon}$. Neighborhood complexity is a notion of sparsity theory, and the theorem extends a classical bound for nowhere dense classes to a much larger, model-theoretically defined family. The submission builds on the Sparsity Lectures submission (Lax12), whose graph classes, nowhere denseness and neighborhood complexity it imports and states its theorems over, and contributes the model-theoretic side — non-copying first-order transductions of relational structures, graph transductions, monadic dependence, weak sparseness — together with three theorems relating the two: weakly sparse monadically dependent classes are nowhere dense; the headline theorem; and nowhere dense classes are monadically dependent (Adler–Adler). Because the nowhere-denseness hypotheses and the almost-linear bound predicate are the separately endorsed definitions of Lax12, these statements compose directly with the sparsity theory stated there, and the surface carries the full classical equivalence that on weakly sparse classes, monadic dependence and nowhere denseness coincide.

The proof package discharges the headline theorem via the paper's VC-dimension sparsification argument; the weakly sparse theorem via Mählmann's Ramsey-theoretic extraction of induced subdivided bicliques (thesis, Lemma 13.8) together with a star-crossing transduction of all graphs; and the Adler–Adler direction via uniform quasi-wideness and a semantic locality argument — the deletion specialization of the flip-breakability route, with hereditarily finite rank-bounded local types of decorated balls and a ball-swap back-and-forth system in place of Gaifman's theorem. A transduction of all graphs would shatter arbitrarily large sets; quasi-wide scattering, a local-type pigeonhole, and the swap lemma refute this.

The classical sparsity and Ramsey material the proofs rest on is assumed from upstream submissions, so the dependency is visible in the archive's proof network: uniform quasi-wideness and almost linear neighborhood complexity of nowhere dense classes from Sparsity Lectures (Lax12), which formalizes the lecture notes of Pilipczuk and Siebertz, and Ramsey's theorem for colourings of pairs with its order-type form for tuples from Finite Ramsey (Lax14). The terminal step of the headline proof composes the statements of the two halves of the paper's Corollary 6 — the weakly sparse theorem stated here and the nowhere dense counting theorem stated in Lax12. What each proof reports beyond Lean's standard logical axioms is exactly the list in its $assumptions$ block.