More about sparse halves in triangle-free graphs

One of Erdős’s conjectures states that every triangle-free graph on vertices has an induced subgraph on vertices with at most edges. We report several partial results towards this conjecture. In particular, we establish the new bound on the number of edges in the general case. We completely prove the conjecture for graphs of girth , for graphs with independence number and for strongly regular graphs. Each of these three classes includes both known (conjectured) extremal configurations, the 5-cycle and the Petersen graph. Bibliography: 21 titles.