What are some ways to prove limits to chromatic numbers, or what are some facts that I could when dealing with large chromatic numbers, such as in the question:
Let G be a graph whose odd cycles are pairwise intersecting, meaning that every two odd cycles in G have a common vertex. Prove that $\chi(G) \leq 5$
Where $\chi(G)$ is G's chromatic number.
I know that odd cycles have chromatic number 3, so any graph containing an odd cycle must have $\chi(G) \geq 3$. I also know that a clique $\kappa_k$ has chromatic number k, but that if $k \gt 3$, then there are at least 2 odd cycles that don't share a vertex.
Are there any other facts about chromatic number that could help me prove this limit?