A DYNAMICAL ARGUMENT FOR A RAMSEY PROPERTY

The paper defines the opposite-Ramsey number r(p,q) as the largest k such that every p-edge-coloring of K_q contains a monochromatic K_k, and asserts there exists q with q(log n) super-polynomial and liminf_{n→∞} r((2n+1)^2, q(n)) < ∞ (Theorem 2.1) . The model observes that for fixed k≥3, the multicolor Ramsey number R_p(k) is polynomial in p up to a logarithmic factor (e.g., R_p(3)=Θ(p log p)), so for p=(2n+1)^2 one has R_p(k) ≤ poly(n)·log n. Since q(log n) being super-polynomial implies q(n) eventually dominates every polynomial in n, one obtains q(n) > R_{(2n+1)^2}(k) for all large n and every fixed k, hence r((2n+1)^2, q(n)) ≥ k for all large n and thus liminf r((2n+1)^2, q(n)) = ∞. This directly contradicts the paper’s main claim. The paper’s dynamical argument constructs, for each n, a specific (2n+1)^2-coloring of K_{S(α^{-n})}, then argues by contradiction that if liminf r→∞, one would obtain arbitrarily large finite 1/(4α)-separated subsets of a compact space, which is impossible; hence the liminf must be finite . However, the combinatorial upper bounds force that very liminf to be infinite for any q with q(log n) super-polynomial, so the theorem cannot hold. Additionally, there is an apparent miscitation: the text attributes the existence of an expansive Z^2-action on T^∞ to reference [2], which (as listed) concerns nonexistence on graphs, not such a construction .