The definitions:
Let $n \in \mathbb{Z}^+$. A positive integer $L$ is called $n$ -magnificent if in every partition of $\{ 1, ... , L \}$ into $n$ non-empty parts, at least one part contains an arithmetic triple (i.e an arithmetic progression whose length is 3). $S(n)$ is defined as the smallest number among the $n$-magnificent.
The problem:
Find the (explicit or implicit) formula, if any, of $S(n)$ with respect to $n$.
$\text{ }$
This is not an assignment. I tried to find some beginning values with computer's help. What I've found is $S(1) = 3$, $S(2) = 9$ and $S(3) = 27$. These results can be wrong, though.
Thanks in advance!
You are looking at the Van der Waerden numbers $W(3,n)$. Your values are correct; the next one is $W(3,4)=76$, and the ones after that are unknown, though the lower bounds $W(3,5)>170$ and $W(3,6)>223$ are known. According to Wolfram MathWorld it is known that there is a constant $c$ such that
$$W(3,n)\le e^{n^c}$$
for all $n$.