  One of Erd\H{o}s's conjectures states that every triangle-free graph on $n$
vertices has an induced subgraph on $n/2$ vertices with at most $n^2/50$ edges.
We report several new partial results towards this conjecture, including the
bound $\frac{27}{1024}n^2$ improving on the best previously known bound
$\frac{n^2}{36}$ due to Krivelevich.
