On a problem of Sivaraman and a problem of Gyárfás

The $\mathit{\text{girth}}$ of a graph $G$, denoted $\mathrm{g}(G)$, is the length of a shortest cycle in $G$. If $G$ contains no cycle, we define $\mathrm{g}(G)=\mathrm{\infty }$. Sivaraman (2020) asked for the optimal $\chi$-bounding function for the class of graphs whose complements have girth at least $6$. Let $F(s)=\max \{\chi (G):\omega (G)\le s,\mathrm{g}(\bar{G})\ge 6\}$. We prove that there exists a constant $c>0$ such that $$c{\left(\frac{s}{\log s}\right)}^{4\mathrm{/}3}\le F(s)\le (1+o(1))\frac{s^{3\mathrm{/}2}}{\log s}\mathrm{.}$$ For small values, we establish the exact results $$F(1)=1,F(2)=2,F(3)=4,F(4)=5,F(5)=6,F(6)=8,$$ and each bound is sharp. A graph $G$ is \emph{almost perfect} if every induced subgraph $H$ of $G$ satisfies $\alpha (H)\omega (H)+1\ge \mathrm{\mid }V(H)\mathrm{\mid }$. Gyárfás (2023) asked whether almost perfect graphs are $\chi$-bounded by the function $g(x)=x+1$. We answer this question in the negative by showing that there is no constant $c$ such that every almost perfect graph $G$ satisfies $\chi (G)\le \omega (G)+c$.