Sparse halves in dense triangle-free graphs

Erd\H{o}s conjectured that every triangle-free graph $G$ on $n$ vertices contains a set of $\lfloor n\mathrm{/}2\rfloor$ vertices that spans at most $n^2\mathrm{/}50$ edges. Krivelevich proved the conjecture for graphs with minimum degree at least $\frac{2}{5}n$. Keevash and Sudakov improved this result to graphs with average degree at least $\frac{2}{5}n$. We strengthen these results by showing that the conjecture holds for graphs with minimum degree at least $\frac{5}{14}n$ and for graphs with average degree at least $(\frac{2}{5} - \varepsilon)n$forsomeabsolute$\varepsilon >0$. Moreover, we show that the conjecture is true for graphs which are close to the Petersen graph in edit distance.