Perfect Matchings in the Semirandom Graph Process

The semirandom graph process is a single player game in which the initially presented an empty on $n$ vertices. In each round, vertex $u$ to independently and uniformly at random. then adaptively selects $v$ adds edge $uv$ graph. For fixed monotone property, objective of force satisfy this property with high probability as few rounds possible. We focus problem constructing perfect matching particular, we present adaptive strategy for achieves $\beta n$ rounds, where value < 1.206$ derived from solution some system differential equations. This improves upon previously best known upper bound $(1+2/e+o(1)) \, n 1.736 rounds. also improve lower $(\ln 2 + o(1)) > 0.693 show that cannot achieve desired less than $\alpha 0.932$ another As result, gap between bounds decreased roughly four times.