关于离散立方体的一个有偏边等周不等式
On a biased edge isoperimetric inequality for the discrete cube
摘要：
离散立方体的“满”边等周不等式（由Harper、Bernstein、Lindsay和Hart提出）指定了一个集合$A \subset \{0,1\}^n$的边边界$\partial A$的最小尺寸，作为$|A|$的函数。一个较弱的（但更广泛使用的）下界是$|\partial A| \geq |A| \log_2(2^n/|A|)$，其中等式成立当且仅当$A$是子立方体。在2011年，第一作者获得了后一个结果的尖锐的“稳定性”版本，证明了如果$|\partial A| \leq |A| (\log(2^n/|A|)+ε)$，则存在子立方体$C$使得$|A ΔC|/|A| = O(ε/\log(1/ε))$。边等周不等式的“弱”版本对于离散立方体上的“$p$ -偏置”测度$μ_p$具有以下众所周知的推广：如果$p \leq 1/2$，或者如果$0 < p < 1$和$A$是单调递增的，则$pμ_p(\partial A) \geq μ_p(A) \log_p(μ_p(A))$。在本文中，我们证明了后一个结果的一个尖锐的稳定性版本，它推广了第一作者的上述结果。也就是说，我们证明了，如果$pμ_p(\partial A) \leq μ_p(A) (\log_p(μ_p(A))+ε)$，那么存在一个子立方体$C$，使得$μ_p(A ΔC)/μ_p(A) = O(ε' /\log(1/ε'))$，其中$ε' =ε\ln (1/p)$。这一结果是一个核心组成部分，在最近的工作证明尖锐的稳定版本的一些Erdens-Ko-Rado型定理极值组合，包括开创性的'完全相交定理'的Ahlswede和Khachatrian。此外，我们证明了一个有偏的措施模拟的“满”边等周不等式，单调递增的集合，我们观察到，这样的模拟不适用于任意集，因此回答了一个问题Kalai。我们使用这个结果给一个新的证明的“充分”边等周不等式，一个依赖于Kruskal-Katona定理。
英文摘要：
The `full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary $\partial A$ of a set $A \subset \{0,1\}^n$, as a function of $|A|$. A weaker (but more widely-used) lower bound is $|\partial A| \geq |A| \log_2(2^n/|A|)$, where equality holds iff $A$ is a subcube. In 2011, the first author obtained a sharp `stability' version of the latter result, proving that if $|\partial A| \leq |A| (\log(2^n/|A|)+ε)$, then there exists a subcube $C$ such that $|A ΔC|/|A| = O(ε/\log(1/ε))$. The `weak' version of the edge isoperimetric inequality has the following well-known generalization for the `$p$-biased' measure $μ_p$ on the discrete cube: if $p \leq 1/2$, or if $0 < p < 1$ and $A$ is monotone increasing, then $pμ_p(\partial A) \geq μ_p(A) \log_p(μ_p(A))$. In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if $pμ_p(\partial A) \leq μ_p(A) (\log_p(μ_p(A))+ε)$, then there exists a subcube $C$ such that $μ_p(A ΔC)/μ_p(A) = O(ε' /\log(1/ε'))$, where $ε' =ε\ln (1/p)$. This result is a central component in recent work of the authors proving sharp stability versions of a number of Erdős-Ko-Rado type theorems in extremal combinatorics, including the seminal `complete intersection theorem' of Ahlswede and Khachatrian. In addition, we prove a biased-measure analogue of the `full' edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the `full' edge isoperimetric inequality, one relying on the Kruskal-Katona theorem.
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