Finding bipartite subgraphs efficiently

Polynomial algorithms are given for the following two problems: given a graph with $n$ vertices and $m$ edges, where $m\ge 3n^{3\mathrm{/}2}$, find a complete balanced bipartite subgraph with parts about $\ln n\mathrm{/}(\ln (n^2\mathrm{/}m))$, given a graph with $n$ vertices, find a decomposition of its edges into complete balanced bipartite graphs having altogether $O(n^2\mathrm{/}\ln n)$ vertices. Previous proofs of the existence of such objects, due to K\H{o}v'ari-S'os-Tur'an, Chung-Erd\H{o}s-Spencer, Bublitz and Tuza were non-constructive.