Highly connected monochromatic subgraphs
We conjecture that for n > 4 (k - 1) every 2-coloring of the edges of the complete graph Kn contains a k-connected monochromatic subgraph with at least n - 2 (k - 1) vertices. This conjecture, if true, is best possible. Here we prove it for k = 2, and show how to reduce it to the case n < 7 k - 6. We prove the following result as well: for n > 16 k every 2-colored Kn contains a k-connected monochromatic subgraph with at least n - 12 k vertices. © 2007 Elsevier B.V. All rights reserved.
Bollobás, B., & Gyárfás, A. (2008). Highly connected monochromatic subgraphs. Discrete Mathematics, 308 (9), 1722-1725. https://doi.org/10.1016/j.disc.2006.01.030