OBJECTIVE AND SOURCE REVIEW
Erdős #627; independent attempt, 2026-09-23. Let $f(n)=\max_{|V(G)|=n}\chi(G)/\omega(G)$. Does $f(n)(\log_2 n)^2/n$ converge? I read all of
627.tex, including the inverse-threshold proof. Its sufficient condition $L=M$ is not a proved necessary condition; calling that equality the exact remaining obstruction overstates the deduction. I test whether elementary regularity of $f$ could close the gap.
WORK: BOUNDED INCREMENTS DO NOT FORCE A LIMIT
For $n\ge1$, \[
0\le f(n+1)-f(n)\le1.
\] The lower bound follows by adding an isolated vertex. For the upper bound, take an optimizing graph $G$ on $n+1$ vertices and delete any vertex, obtaining $H$. Then $\chi(G)\le\chi(H)+1$ and $\omega(G)\ge\omega(H)\ge1$, so $\chi(G)/\omega(G)\le f(n)+1$. Nevertheless these inequalities plus $f(n)=\Theta(n/(\log_2 n)^2)$ do not imply convergence. To see this, for fixed $a>b>0$ define for large real $x$ \[
h(x)=\frac{x}{(\log_2 x)^2}\left(a+b\sin(\log\log x)\right),
\qquad F(n)=\lfloor h(n)\rfloor.
\] Differentiation gives $h'(x)=(a+b\sin(\log\log x))/(\log_2 x)^2+O((\log x)^{-3})$. Thus eventually $0<h'(x)<1$, and $F$ is nondecreasing with increments zero or one. Yet \[
\liminf F(n)(\log_2n)^2/n=a-b,\qquad
\limsup F(n)(\log_2n)^2/n=a+b.
\] Integer rounding contributes only $o(1)$, and the phases of $\log\log n$ approach the maxima and minima along suitable integer subsequences.
ADVERSARIAL VERIFICATION AND REMAINING GAP
The constructed $F$ is an abstract sequence, not the graph extremal function. It disproves only a proposed inference from monotonicity, bounded jumps and order of growth. The source's inverse transformation remains valid, but transfers rather than resolves oscillation. Its inequalities $\liminf g\ge L^2$ and $\limsup g=M^2$ allow convergence even when $L<M$; a stronger graph or Ramsey argument is still needed.
SOURCES CHECKED
Complete local source
gpt_pro_5.4/627.tex; primary Araujo--Filipe--Miyazaki record
https://arxiv.org/abs/2512.16062, checked 2026-09-23. No unverified numerical constant from that paper is needed above.
FINAL
UNRESOLVED. Elementary regularity is established and shown insufficient to force the desired limit.
COMPLETION ESTIMATE
Conservative subjective progress toward convergence.
COMPLETION: 8%