  In this paper we solve two problems of Esperet, Kang and Thomasse as well as
Li concerning (i) induced bipartite subgraphs in triangle-free graphs and (ii)
van der Waerden numbers. Each time random greedy algorithms allow us to go
beyond the Lovasz Local Lemma or alteration method used in previous work,
illustrating the power of the algorithmic approach to the probabilistic method.
