  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 $k \ge
16$.
