DIMENSION BOUNDS FOR SYSTEMS OF EQUATIONS WITH GRAPH STRUCTURE

The paper’s main result (Theorem 17) states that for a d-compatible set X and an induced forest F without isolated vertices, choosing L as all leaves except one per component yields x_{L ∪ (G \ F)} ⇒_{d^{|F\L|}} x_{F\L} and hence dim_c(X) ≤ |G| − |F| + |L|; in the strongly d-compatible case the no-isolated-vertices hypothesis can be dropped . The authors reduce Theorem 17 to the “pure forest” case (Theorem 18) via relative compatibility of induced subgraphs (Lemma 12) , combine components using the monotonicity and conjunction properties of finite determinacy (Lemma 6) , and prove the tree step using a rooted-tree order (Tree Lemma 20) , . The candidate’s solution gives a direct peeling proof on the induced forest using a “boundary leaf” lemma and iterated d-determinacy along edges, obtaining the same exponent d^{|F\L|} and the same dimension bound. This is logically correct and equivalent in conclusion to Theorem 17, though presented differently. The paper also notes a stronger form in the strong-compatibility setting by including isolated vertices Z into F and obtaining dim_c(X) ≤ |G| − |F| + l(F) − c(F) − |Z| (eq. (9)); the candidate’s write-up does not emphasize this strengthening but remains valid and consistent with Theorem 17 as stated .