The Multicolour Ratio $R_k(C_2n+1)/R_k(K_3)$ Vanishes

For each fixed $n 1$, the ratio of the $k$-colour Ramsey numbers of the odd cycle $C_2n+1$ and of the triangle tends to zero as the number of colours grows. The proof is a comparison of exponential bases, and the whole subtlety is in where the cycle length is allowed to appear. Bondy and Erdős bound the numerator by $(2n+1)2^k$ and Schur bounds the denominator below by $3.199^k$, so the ratio is at most $(2n+1)(2/3.199)^k$. The base $2/3.199$ is below one and does not depend on $n$; the entire dependence on the cycle length sits in the constant, where a geometric limit does not see it. A comparison in which $n$ enters the base instead does not vanish at all: the base $(4n-2)/3.199$ exceeds one from $n = 2$.