Using Erdős’s Methods to Study Yorke’s Problems

The paper states Theorem 3.3 and attempts to prove that for all d ≥ 4, k > 1, the relative dimensions |GA_{k+1,d}|/|GA_{k,d}|, |GA_{k+1,d}|/|GA_{2,d}|, and |GA_{k,d}|/|GA_{1,d}| are infinite. The setup and definitions for GA_{k,d}, P_n GA_{k,d}, and the "star representation" are clear (definition of the relative dimension and Lemma 3.2’s role are explicitly given). The lemma claims that for d ≥ 4, forgetting colors yields all trees of degree k+1 on n vertices; this is asserted via Proposition 2.5 but not actually constructed (no concrete composition argument is provided) . The main proof then replaces the relevant class of degree-bounded unlabeled trees by “k-ary (uncolored) trees with n vertices,” invoking the Catalan-type formula N(k) = 1/((k−1)n+1) binom(kn,n), and derives a lower bound |P_n GA_{k,d}|/|P_n GA_{k-1,d}| ≥ N(k+1)/(3^n N(k)) via Stirling’s approximation . However: (i) the identification of the uncolored class arising from GA_{k,d} with “k-ary trees” is not justified and conflicts with the earlier claim (all trees of maximum degree ≤ k+1), and (ii) the subsequent asymptotic inequality and limit evaluation are arithmetically incorrect: the displayed lower bound behaves like ((k+1)/k)^{2n} divided by an exponential factor ≥ 3^n (and further multiplied by (1−1/k^2)^{(k−1)n}<1), so the right-hand side actually decays to 0 for fixed k≥2, not ∞. Thus the paper’s proof of Theorem 3.3 is flawed. By contrast, the model’s solution gives a correct lower-bounding reduction to degree-constrained unlabeled (Pólya) trees together with standard DLW asymptotics and monotonicity of the growth constant, which rigorously yields exponential divergence of the ratio and hence the theorem.