Existence and value of the chromatic-to-clique ratio limit

Determine whether the limit \(\lim_{n\to\infty} f(n)/(n/(\log n)^2)\) exists and, if it exists, determine its value, where \(f(n)\) is the maximum of \(\chi(G)/\omega(G)\) over all graphs \(G\) on \(n\) vertices.

Background

For each nn, the function f(n)f(n) measures the largest possible ratio between chromatic number and clique number among graphs with nn vertices. Erdős established that f(n)=Θ(n/(log⁡n)2)f(n)=\Theta(n/(\log n)^2), but the paper records the unresolved question of whether the normalized quantity has a limiting constant.

The paper proves improved upper bounds for this normalized ratio and relates its asymptotic behavior to Ramsey-number growth, but does not establish existence of the limit.

References

Erd\H{o}s showed that f(n) = \Theta(n/(\log n)2) and asked whether the following limit exists: \begin{equation}\label{limit2} \lim_{n \to \infty} \frac{f(n)}{n/(\log n)2}. \end{equation}

limit2:

lim⁡n→∞f(n)n/(log⁡n)2.\lim_{n \to \infty} \frac{f(n)}{n/(\log n)^2}.

— On the maximum ratio between chromatic number and clique number  (2512.16062 - Araujo et al., 18 Dec 2025) in Section 1, Introduction