Independent set size in triangle-free graphs
If you don't care about the constants, then the answer is 'yes' by an easy reduction:

Take a triangle-free graph $G$ with average degree $d_{avg}$. Create a new graph $G'$ which is obtained from $G$ by deleting all vertices with degree $\ge 2d_{avg}$. Obviously, any independent set in $G'$ is an independent set in $G$, and $G'$ is triangle-free. Also, the number of vertices in $G'$ is at least $n(G)/2$. Now we can apply the Alon--Spencer proof.