Locally bipartite subgraphs via multicolor Ramsey numbers

A famous conjecture of Erd\H{o}s and Hajnal (1969) states that for every integer $g\ge 4$ there is a smallest function $f_g:\mathbb{N}\to \mathbb{N}$ such that every graph of chromatic number at least $f_g(k)$ contains a subgraph of chromatic number $k$ and girth at least $g$. So far, this has only been proved for $g=4$ by R"odl (1977), with $f_4(k)$ bounded by a tower of height $\mathrm{\Theta }(k^2\log k)$. We exhibit a surprising connection between finding high-chromatic subgraphs of large odd-girth (avoiding short odd cycles) and lower-bounding multicolor Ramsey numbers of odd cycles. Using this connection, we prove that for every odd $g\ge 5$ there is a function $h_g:\mathbb{N}\to \mathbb{N}$ growing as a power tower of height $\frac{g-3}{2}$ such that every graph of chromatic number at least $h_g(k)$ contains a subgraph of chromatic number at least $k$ and odd-girth at least $g$. This proves a conjecture of Mohar and Wu (2018), addresses a question of Erd\H{o}s and Hajnal (1975), and for $g=5$ improves R"odl’s bound on $f_4(k)$ to a single-exponential. We extend this to a much more general meta-theorem which applies to many graph parameters: if $f$ is the fractional chromatic number, the Hall ratio, or the strict vector chromatic number (Lov'{a}sz-Theta-function of the complement), then for every $k,g\in \mathbb{N}$, every graph $G$ with sufficiently large $f(G)$ contains a subgraph $G^{\mathrm{\prime }}$ of odd-girth at least $g$ with $f(G^{\mathrm{\prime }})\ge k$. The key Ramsey-theoretic ingredient is a new lower bound on Ramsey numbers of odd cycles. For $p\ge 1$, let ${\mathcal{O}}_p=\{C_3,C_5,\dots ,C_{2p+1}\}$. We show that $R_k({\mathcal{O}}_p)\ge ({\log }^{(p-1)}k{)}^{k\mathrm{/}3-o(k)}$ for every fixed $p$, where ${\log }^{(p-1)}$ denotes the $(p-1)$-fold iterated logarithm. This yields the first superexponential lower bound on multicolor Ramsey numbers of fixed odd cycles, and extends the recent breakthrough by OpenAI for triangles.