Consider the following semi-random multigraph process which starts with an empty graph on $n$ vertices. In every round of the process, one vertex $v$ of the graph is picked uniformly at random and independently of all previous rounds. We then choose an additional vertex (according to a strategy of our choice) and connect it by an edge to $v$.
For various natural monotone increasing graph properties $\mathcal{P}$, we prove tight upper and lower bounds on the minimum (extended over the set of all possible strategies) number of rounds required by the process to obtain, with high probability, a graph that satisfies $\mathcal{P}$. Along the way, we show that the process is general enough to approximate (using suitable strategies) several well-studied random graph models.
Joint work with Omri Ben-Eliezer, Gal Kronenberg, Olaf Parczyk, Clara Shikhelman and Miloš Stojaković.