Perfect matchings and loose Hamilton cycles in the semirandom hypergraph model

We study the 2-offer semirandom 3-uniform hypergraph model on n vertices. At each step, we are presented with 2 uniformly random vertices. We choose any other vertex, thus creating a hyperedge of size 3. We show a strategy that constructs a perfect matching, and another that constructs a loose Hamilton cycle, both succeeding asymptotically almost surely within Θ(n) steps. Both results extend to s-uniform hypergraphs. Our methods are qualitatively different from those that have been used for semirandom graphs. Much of the analysis is done on an auxiliary graph that is a uniform k-out subgraph of a random bipartite graph, and this tool may be useful in other contexts.