The site's "Let $\epsilon>0$" admits every positive $\epsilon$, and for $\epsilon\ge2$ the first question fails trivially: for even $k$ the factor $(1-\epsilon)^k$ is at least $1$, and $G=K_k$ has $\chi(G)=k$ and $R(G)=R(k)\le(1-\epsilon)^kR(k)$, so the inequality fails for every even $k$. The smallest instance is $\epsilon=2$, where $(1-\epsilon)^k=1$ for every even $k$. The check is elementary and was made here; the formal-conjectures statement already restricts $\epsilon$ to $(0,1)$, its docstring noting that the restriction excludes negative bases.

The change replaces "Let $\epsilon>0$" by "Let $0<\epsilon<1$"; nothing else changes. The evidence is the poser's own words. Erdős [Er95], Section II.15, p. 14 of the typescript (its card), writes that "$r(G,G)>(1-\varepsilon)^nr(n)$ should hold for some $0<\varepsilon<1$", so Erdős's $\varepsilon$ lies in $(0,1)$. The site's commentary agrees: its remark "Since $R(k)\leq 4^k$ this is trivial for $\epsilon\geq 3/4$" rests on $(1-\epsilon)^k4^k\le1$, which holds for every $k$ only when $3/4\le\epsilon\le5/4$, so the remark is true only when $\epsilon$ is bounded and does not contemplate the trivially false $\epsilon\ge2$; the formal-conjectures statement, which counts with the site, takes $0<\epsilon<1$. No text of the poser lets $\epsilon$ exceed $1$, so the defect is the site's. The site's universal question against Erdős's "for some" is the site's own restatement and stands; Formulation records Erdős's wording with its answer. The change moves no standing: both questions of the corrected Statement are open, and no result about the site's wording beyond the trivial failure above is recorded.