On the local density problem for graphs of given odd-girth

Erd\H{o}s conjectured that every $n$-vertex triangle-free graph contains a subset of $\lfloor n\mathrm{/}2\rfloor$ vertices that spans at most $n^2\mathrm{/}50$ edges. Extending a recent result of Norin and Yepremyan, we confirm this conjecture for graphs homomorphic to so-called Andr'asfai graphs. As a consequence, Erd\H{o}s’ conjecture holds for every triangle-free graph $G$ with minimum degree $\delta (G)>10n\mathrm{/}29$ and if $\chi (G)\le 3$ the degree condition can be relaxed to $\delta (G)>n\mathrm{/}3$. In fact, we obtain a more general result for graphs of higher odd-girth.