Forbidden Properly Edge-Colored Subgraphs that Force Large Highly Connected Monochromatic Subgraphs

We consider the connected graphs G that satisfy the following property: If $$n \gg m \gg k$$n≫m≫k are integers, then any coloring of the edges of $$K_{n}$$Kn, using m colors, containing no properly colored copy of G, contains a monochromatic k-connected subgraph of order at least $$n - f(G, k, m)$$n-f(G,k,m) where f does not depend on n. If we let $$\mathscr {G}$$G denote the set of graphs satisfying this statement, we exhibit some infinite families of graphs in $$\mathscr {G}$$G as well as conjecture that the cycles in $$\mathscr {G}$$G are precisely those whose lengths are divisible by 3. Our main result is that $$C_{6} \in \mathscr {G}$$C6∈G.