Sparse spanning $k$-connected subgraphs in tournaments
  In 2009, Bang-Jensen asked whether there exists a function $g(k)$ such that every strongly $k$-connected $n$-vertex tournament contains a strongly $k$-connected spanning subgraph with at most $kn + g(k)$ arcs. In this paper, we answer the question by showing that every strongly $k$-connected $n$-vertex tournament contains a strongly $k$-connected spanning subgraph with at most $kn + 750k^2\log(k+1)$ arcs. 
Dong Yeap Kang, Jaehoon Kim, Younjin Kim, Geewon Suh
math.CO