Subgraph Games in the Semi-Random Graph Process and Its Generalization to Hypergraphs

The semi-random graph process is a single-player game that begins with an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$ and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. We focus on the problem of constructing a subgraph isomorphic to an arbitrary, fixed graph $G$. Let $\omega = \omega(n)$ be any function tending to infinity as $n \to \infty$. In (Omri Ben-Eliezer et al. "Semi-random graph process". In: Random Structures & Algorithms 56.3 (2020), pp. 648-675) it was proved that asymptotically almost surely one can construct $G$ in less than $n^{(d-1)/d} \omega$ rounds where $d \ge 2$ is the degeneracy of $G$. It was also proved that the result is sharp for $G = K_{d+1}$, that is, asymptotically almost surely it takes at least $n^{(d-1)/d} / \omega$ rounds to create $K_{d+1}$. Moreover, the authors conjectured that their general upper bound is sharp for all graphs $G$. We prove this conjecture here. We also consider a natural generalization of the process to $s$-uniform hypergraphs, the semi-random hypergraph process in which $r \ge 1$ vertices are presented at random, and the player then selects $s-r \ge 1$ vertices to form an edge of size~$s$. Our results for graphs easily generalize to hypergraphs when $r=1$; the threshold for constructing a fixed $s$-uniform hypergraph $G$ is, again, determined by the degeneracy of $G$. However, new challenges are mounting when $r \ge 2$; thresholds are not even known for complete hypergraphs. We provide bounds for this family and determine thresholds for some sparser hypergraphs.