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