Highly connected monochromatic subgraphs of two-colored complete graphs

We show that each 2-edge coloring of the complete graph on n vertices leads to a monochromatic k  -connected subgraph on at least n−2(k−1)n−2(k−1) vertices, provided n≥4k−3n≥4k−3. It settles in the affirmative a conjecture of Bollobás and Gyárfás.