Suppose we have a graph $G$ with $n$ vertices. If we color the edges of $G$ by two colors, then we can conclude by Konig's Theorem that there exists a monochromatic subtree $T$ with at least $n/{\alpha}$ vertices, $\alpha$ is the size of maximum independent set in the graph. 


I was wondering if there is any theorem that I can conclude there exists an INDUCED monochromatic subtree, not just subtree?


Thanks in advance for any help/comment!



Induced monochromatic subtree in a graph which is colored by two colors
Every connected $n$-vertex graph with $O(n)$ edges has an induced tree of size $2\log\log n+O(\log\log\log n)$ (I guess the constants in the $O$'s depend on each other); see


P. Erdős, M. Saks, and V. T. Sós. Maximum induced trees in graphs. J. Combinatorial Theory, Series B, 41(1):61 – 79, 1986. doi:10.1016/0095-8956(86)90028-6


Additionally, connected triangle-free graphs have induced trees of size $\sqrt n$ and connected clique-free graphs for larger clique sizes have logarithmic induced trees; see


J. Fox, P.-S. Loh, and B. Sudakov. Large induced trees in $K\_r$-free graphs. J. Combinatorial Theory, Series B, 99(2):494 – 501, 2009. doi:10.1016/j.jctb.2008.10.001


So if you know something about the connectivity and subgraphs of the two colors then you can conclude something about the sizes of the monochromatic trees. 


