  We describe two constructions of (very) dense graphs which are edge disjoint
unions of large {\em induced} matchings. The first construction exhibits graphs
on $N$ vertices with ${N \choose 2}-o(N^2)$ 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 $N^{2-\delta}$
induced matchings, where $\delta > 0.058$. 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 H{\aa}stad 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.
