Monochromatic Connectivity in Monochromatic-star-free Graphs

We consider edge-colorings of complete graphs in which each color induces a subgraph that does not contain an induced copy of \( K_{1,t} \), for some \( t \geq 3 \). It turns out that such colorings, if the underlying graph is sufficiently large, contain spanning monochromatic \( k \)-connected subgraphs. Furthermore, there exists a color, say blue, such that every vertex has very few incident edges in colors other than blue.