About independent set in triangle free graphs From Wikipedia :


  An independent set of $\sqrt{n}$ vertices in an $n$-vertex triangle-free graph is easy to find: either there is a vertex with more than $\sqrt{n}$ neighbors (in which case those neighbors are an independent set) or all vertices have less than $\sqrt{n}$ neighbors (in which case any maximal independent set must have at least $\sqrt{n}$ vertices)


Why does any maximal stable set have at least square root of $n$ vertices?

Thanks.
The point is that the graph is triangle free, so if $j$ and $k$ are both neighbours of $i$, then the edge $\{j,k\}$ is not in the graph.  This means that $N_i = \{j : \text{$i$ is a neighbour of $j$}\}$ is an independent set for any $i$.

The other fact being used is that a graph with $n$ vertices of maximum degree $\Delta$ has an independent set of size at least $\lfloor n/(\Delta+1)\rfloor$.