IMC 2022 · Problem 4

Let $n > 3$ be an integer. Let $\Omega$ be the set of all triples of distinct elements of $\{1, 2, \ldots, n\}$. Let $m$ denote the minimal number of colours which suffice to colour $\Omega$ so that whenever $1 \le a < b < c < d \le n$, the triples $\{a, b, c\}$ and $\{b, c, d\}$ have different colours. Prove that

$$\frac{1}{100} \log\log n \leqslant m \leqslant 100 \log\log n.$$