Fractional Erdős–Hajnal conjecture for subgraphs of prescribed girth

Let $x\ge1$ be real and let $g\ge3$ be an integer. For a graph $G$, let $\chi_f(G)$ denote its fractional chromatic number, and let the girth of a graph be the length of its shortest cycle. Fractional Erdős–Hajnal conjecture. For every real number $x\ge1$ and any integer $g\ge3$, there exists a positive number $k(x,g)$ such that every graph $G$ with $\chi_f(G)\ge k(x,g)$ contains a subgraph $H$ of girth at least $g$ and with $\chi_f(H)\ge x$. This extends the paper’s proved triangle-free case, corresponding to $g=4$, to arbitrary prescribed girth; the proposed general statement remains open.