Extremal bipartite independence number and balanced coloring

In this paper, we establish a couple of results on extremal problems in bipartite graphs. Firstly, show that every sufficiently large graph with average degree $D$ and $n$ vertices each side has balanced independent set containing $(1-\epsilon) \frac{\log D}{D} n$ from for small $\epsilon > 0$. Secondly, prove the vertex maximum at most $\Delta$ can be partitioned into $(1+\epsilon)\frac{\Delta}{\log \Delta}$ sets. Both these are algorithmic best possible up to factor 2, which might hard improve as evidenced by phenomenon known `algorithmic barrier' literature. The first result improves recent theorem Axenovich, Sereni, Snyder, Weber slightly more general setting. second Feige Kogan about coloring