About independent set in triangle free graphs
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 $\lfloor n/(\Delta+1)\rfloor$.