Bollob\'{a}s and Gy\'{a}rf\'{a}s conjectured that for any k,nβZ+ with n>4(kβ1), every 2-edge-coloring of the complete graph on
n vertices leads to a k-connected monochromatic subgraph with at least
nβ2k+2 vertices. We find a counterexample with n=5kβ2β2kβ1βββ3, thus disproving the conjecture, and we show the
conjecture is true for nβ₯5kβmin{4kβ2β+3,0.5k+4}