Sparse halves in dense triangle-free graphs

Erdős [3] conjectured that every triangle-free graph G on n   vertices contains a set of ⌊n/2⌋⌊n/2⌋ vertices that spans at most n2/50n2/50 edges. Krivelevich proved the conjecture for graphs with minimum degree at least 25n[9]. In [8] Keevash and Sudakov improved this result to graphs with average degree at least 25n. We strengthen these results by showing that the conjecture holds for graphs with minimum degree at least 514n and for graphs with average degree at least (25−γ)n for some absolute γ>0γ>0. Moreover, we show that the conjecture is true for graphs which are close to the Petersen graph in edit distance.