Highly connected monochromatic subgraphs

We conjecture that for n>4(k-1)n>4(k-1) every 2-coloring of the edges of the complete graph KnKn contains a k  -connected monochromatic subgraph with at least n-2(k-1)n-2(k-1) vertices. This conjecture, if true, is best possible. Here we prove it for k=2k=2, and show how to reduce it to the case n<7k-6n<7k-6. We prove the following result as well: for n>16kn>16k every 2-colored KnKn contains a k  -connected monochromatic subgraph with at least n-12kn-12k vertices.