The semi-random graph process is a single player game in which the player is initially presented 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. In this paper, we investigate the following three properties: containing a complete graph of order $k$, having the chromatic number at least $k$, and not having an independent set of size at least $k$.

翻译：半随机图过程是一种单人游戏，其中玩家初始面对一个包含 $n$ 个顶点的空图。每轮中，一个顶点 $u$ 独立且均匀随机地呈现给玩家。玩家随后自适应地选择一个顶点 $v$，并将边 $uv$ 添加到图中。对于固定的单调图性质，玩家的目标是在尽可能少的轮数内，以高概率迫使图满足该性质。本文研究了以下三种性质：包含一个 $k$ 阶完全图、色数至少为 $k$、以及不存在一个大小至少为 $k$ 的独立集。