Sparse halves in dense triangle-free graphs

In this paper we study a conjecture of Erdös that any triangle-free graph G on n vertices should contain a set of ⌊ n / 2 ⌋ vertices that spans at most n 2 / 50 edges. Krivelevich proved the conjecture for graphs with minimum degree at least 2 5 n . Keevash and Sudakov improved this result to graphs with average degree at least 2 5 n . We strengthen these results further by showing that the conjecture holds for graphs with minimum degree at least 5 14 n and average degree at least ( 2 5 − ϵ ) n for some absolute ϵ > 0 .