I have recently learned about Ramsey Theory and I think that sometimes in some olympiads problem it may be a really powerful technique, for example in this problem:
A magician has $66$ potions. Two potions when react can generate $4$ different effect. Prove that there exist $3$ potions $A,B,C$ such that the reactions $A-B , A-C, B-C$ generate the same effect
I thought that if we consider a connected graph with $66$ nodes, the problem is isomorphic to show that there exist a monochromatic triangle in this graph. So the problem is equivalent to show(or at least I think) that: $$R(3,0,0,0)\leq 66\ \ \ \ $$ Are there some useful upper bounds for generalized Ramsey Numbers? :)
There are, yes.
Specifically for Ramsey numbers for monochromatic triangles, the result you need follows directly from the inequality proved here that $$R(\underbrace{3,3,\dots,3,3}_{k+1}) \le (k+1)(R(\underbrace{3,3,\dots,3}_k)-1)+2.$$ Since we know $R(3,3) = 6$, we can apply this inequality once to conclude that $R(3,3,3) \le 3 \cdot 5 + 2 = 17$ and then apply it again to conclude that $R(3,3,3,3) \le 4 \cdot 16 + 2 = 66$.
The number $R(3,3,3,3)$ is exactly the bound needed in this question: if we draw a graph whose vertices are potions, and color each edge according to the reaction produced by the vertices it joins, then we have a $4$-colored complete graph $K_{66}$, and what we're looking for is exactly a monochromatic triangle.