On the number of edges in a graph with no (k+1)-connected subgraphs

Mader proved that for k≥1 and n≥2k, every n-vertex graph with no (k+1)-connected subgraphs has at most (1+12)(n−k) edges. He also conjectured that for n large with respect to k, every such graph has at most 32(k−13)(n−k) edges. Yuster improved Mader's upper bound to 193120k(n−k) for n≥9k4. In this note, we make the next step towards Mader's Conjecture: we improve Yuster's bound to 1912k(n−k) for n≥5k2.