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.
['Noga Alon', 'Ankur Moitra', 'Benny Sudakov']
Nearly complete graphs decomposable into large induced matchings and their applications