The Erdős-Hajnal conjecture for odd-girth

A famous conjecture of Erd\H{o}s and Hajnal from 1969 states that for every integer $g\ge 4$ there exists a (smallest) function $f_g:\mathbb{N}\rightarrow \mathbb{N}$ such that every graph of chromatic number at least $f_g(k)$ contains a subgraph with chromatic number at least $k$ and girth at least $g$. So far, this has only been proved for $g=4$ by R\"odl in 1977 and remains open for every $g\ge 5$. R\"odl's elegant proof yields an upper bound on $f_4(k)$ which is a tower of $k$-s of height $\Theta(k^2\log k)$, suggesting the problem of improving this enormous bound. We deduce a single-exponential bound $$f_4(k)\le e^{k^{3+o(1)}}$$ from OpenAI's recent lower bound on multicolor Ramsey numbers of triangles. Using a generalization of the latter result to multi-color Ramsey numbers of odd cycles from a companion paper, we show that for every odd $g\ge 5$ there is a function $h_g:\mathbb{N}\rightarrow \mathbb{N}$ growing at most as a power tower of height $\frac{g-3}{2}$ such that every graph of chromatic number at least $h_g(k)$ has a subgraph of chromatic number at least $k$ and odd-girth at least $g$. This proves a conjecture of Mohar and Wu from 2018.