Let Y3,2β be the 3-graph with two edges intersecting in two vertices. We
prove that every 3-graph H on n vertices with at least max{(34Ξ±nβ),(3nβ)β(3nβΞ±nβ)}+o(n3)
edges contains a Y3,2β-tiling covering more than 4Ξ±n vertices, for
sufficiently large n and 0<Ξ±<1/4. The bound on the number of edges
is asymptotically best possible and solves a conjecture of the authors for
3-graphs that generalizes the Matching Conjecture of Erd\H{o}s