A Ramsey Type problem for highly connected subgraphs

Bollob\'{a}s and Gy\'{a}rf\'{a}s conjectured that for any \(k, n \in \mathbb{Z}^+\) with \(n > 4(k-1)\), every 2-edge-coloring of the complete graph on \(n\) vertices leads to a \(k\)-connected monochromatic subgraph with at least \(n-2k+2\) vertices. We find a counterexample with \(n = \lfloor 5k-2.5-\sqrt{8k-\frac{31}{4}} \rfloor\), thus disproving the conjecture, and we show the conclusion holds for \(n > 5k-2.5-\sqrt{8k-\frac{31}{4}}\) when \(k \ge 16\).