论文标题
概率直觉可容纳一类小型游戏
Probabilistic intuition holds for a class of small subgraph games
论文作者
论文摘要
Consider the following two-player game on the edges of $K_n$, the complete graph with $n$ vertices: Starting with an empty graph $G$ on the vertex set of $K_n$, in each round the first player chooses $b \in \mathbb{N}$ edges from $K_n$ which have not previously been chosen, and the second player immediately and irrevocably picks one of these edges and adds it to $ g $。 We show that for any graph $H$ with at least one edge, if $b < c n^{1/m(H)}$, where $c = c(H) > 0$ only depends on $H$ and $m(H)$ is the usual density function, then the first player can ensure the resulting graph $G$ contains $Ω(n^{v(H)} / b^{e(H)})$ copies of $H$.除了常数$ c $以外，在$ b $上的界限是最好的，它表明，所得图的密度可以强制执行$ h $的外观，并在erdős-rényi随机图中与阈值相吻合。这解决了Bednarska-Bzdȩga，Hefetz和Luczak的猜想，并提供了一类杰出的游戏类别，概率直觉准确地预测了结果。第一个玩家的策略是确定性的，具有多项式运行时间，其程度取决于$ h $的大小。
Consider the following two-player game on the edges of $K_n$, the complete graph with $n$ vertices: Starting with an empty graph $G$ on the vertex set of $K_n$, in each round the first player chooses $b \in \mathbb{N}$ edges from $K_n$ which have not previously been chosen, and the second player immediately and irrevocably picks one of these edges and adds it to $G$. We show that for any graph $H$ with at least one edge, if $b < c n^{1/m(H)}$, where $c = c(H) > 0$ only depends on $H$ and $m(H)$ is the usual density function, then the first player can ensure the resulting graph $G$ contains $Ω(n^{v(H)} / b^{e(H)})$ copies of $H$. The bound on $b$ is the best possible apart from the constant $c$ and shows that the density of the resulting graph for which it is possible to enforce the appearance of $H$ coincides with a threshold for the appearance in the Erdős-Rényi random graph. This resolves a conjecture by Bednarska-Bzdȩga, Hefetz, and Luczak and provides a prominent class of games for which probabilistic intuition accurately predicts the outcome. The strategy of the first player is deterministic with polynomial running time, with the degree depending on the size of $H$.
