Sufficient average degree conditions for the existence of large highly connected subgraphs

Mader proved that every sufficiently large graph with average degree at least $(2+\sqrt{2})k$ has a $(k+1)$-connected subgraph. He also conjectured that an average degree of at least $3k$ is sufficient. The best known sufficient factor was improved by multiple authors but never reached $3$. In the present paper, it is further improved to $3.109$. In addition, the obtained $(k+1)$-connected subgraph is constrained to have more than $1.2k$ vertices. Moreover, similar conditions on the average degree are proven to be sufficient for the existence of even greater $(k+1)$-connected subgraphs.