Let $r\ge 4$ be an integer and consider the following game on the complete graph $K_n$ for $n\in r\mathbb{Z}$: Two players, Maker and Breaker, alternately claim previously unclaimed edges of $K_n$ such that in each turn Maker claims one and Breaker claims $b\in \mathbb{N}$ edges. Maker wins if her graph contains a $K_r$-factor, that is a collection of $n/r$ vertex-disjoint copies of $K_r$, and Breaker wins otherwise. In other words, we consider a b-biased $K_r$-factor Maker–Breaker game. We show that the threshold bias for this game is of order $n^{2/(r+2)}$. This makes a step towards determining the threshold bias for making bounded-degree spanning graphs and extends a result of Allen et al. who resolved the case $r\in \{3,4\}$ up to a logarithmic factor.

让 $r\ge 4$ 是一个整数，并在完整图上考虑以下游戏 ${钾}_n$ 为了 $n\in r\mathbb{Z}$: Maker 和 Breaker 两个玩家轮流占据之前无人认领的边缘 ${钾}_n$ 使得在每一轮中 Maker 声称一个而 Breaker 声称 $乙\in \mathbb{N}$边缘。如果 Maker 的图表包含一个${钾}_r$-factor，这是一个集合 $n/r$ 顶点不相交的副本 ${钾}_r$，否则 Breaker 获胜。换句话说，我们考虑 a b -biased${钾}_r$-factor Maker–Breaker 游戏。我们证明了这个游戏的阈值偏差是有序的$n^{2/(r+2)}$. 这朝着确定用于制作有界度跨越图的阈值偏差迈出了一步，并扩展了 Allen 等人的结果。谁解决了这个案子$r\in \{3,4\}$ 直到一个对数因子。