A stronger form was established by Gao, Huo, and Ma [GaHuMa21], who proved that if a graph $G$ has chromatic number $\chi(G)\geq 2k+3$ then $G$ contains cycles of $k+1$ consecutive odd lengths.
He, Ma, and Yang [HeMaYa21] have proved this conjecture when $n=q^2+q+1$ for some even integer $q$.
This was solved in the affirmative if the minimum degree is larger than some absolute constant by Liu and Montgomery [LiMo20] (therefore disproving the above stronger conjecture of Erdős and Gyárfás). Liu and Montgomery prove much stronger result: if the average degree of $G$ is sufficiently large, then there is some large integer $\ell$ such that for every even integer $m\in [(\log \ell)^8,\ell]$, $G$ contains a cycle of length $m$.
Solved by Verstraëte [Ve05], who gave a non-constructive proof that such a set $A$ exists.
Liu and Montgomery [LiMo20] proved that in fact this is true when $A$ is the set of powers of $2$ (more generally any set of even numbers which doesn't grow too quickly) - in particular this contradicts the previous belief of Erdős.
The answer is yes, proved by Gruslys and Letzter [GrLe20].
Even stronger, is there some $c>0$ such that, for all large $k$, $R(G)>cR(k)$ for every graph $G$ with chromatic number $\chi(G)=k$?
Since $R(k)\leq 4^k$ this is trivial for $\epsilon\geq 1/4$. Yuval Wigderson points out that $R(G)\gg 2^{k/2}$ for any $G$ with chromatic number $k$ (via a random colouring), which asymptotically matches the best-known lower bounds for $R(k)$.
Is it true that, for every $\mathcal{F}$, there exists some constant $C_{\mathcal{F}}>0$ and $G\in\mathcal{F}$ such that for all large $n$ \[\mathrm{ex}(n;G)\leq C_{\mathcal{F}}\mathrm{ex}(n;\mathcal{F})?\]
A construction due to Pyber, Rödl, and Szemerédi [PRS95] shows that this is best possible.
See also [483].