More about sparse halves in triangle-free graphs

One of Erdős'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 partial results towards this conjecture. In particular, we establish the new bound $27n^2/1024$ on the number of edges in the general case. We completely prove the conjecture for graphs of girth $\geqslant 5$, for graphs with independence number $\geqslant 2n/5$ and for strongly regular graphs. Each of these three classes includes both known (conjectured) extremal configurations, the 5-cycle and the Petersen graph.
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