We describe two constructions of (very) dense graphs which are edge disjoint unions of large induced matchings. The first construction exhibits graphs on N vertices with ( N 2)-o(N 2) edges, which can be decomposed into pairwise disjoint induced matchings, each of size N 1-o(1). The second construction provides a covering of all edges of the complete graph K N by two graphs, each being the edge disjoint union of at most N 2-δ induced matchings, where δ>0.076. This disproves (in a strong form) a conjecture of Meshulam, substantially improves a result of Birk, Linial and Meshulam on communicating over a shared channel, and (slightly) extends the analysis of Hastad and Wigderson of the graph test of Samorodnitsky and Trevisan for linearity. Additionally, our constructions settle a combinatorial question of Vempala regarding a candidate rounding scheme for the directed Steiner tree problem.