AI 中文总结
本文研究有偏测度下的交叉相交族问题，通过对数赔率插值结合坐标截面归纳、半正定估计等方法，证明了相关尖锐不等式，确认了Suda等人的两个猜想，刻画了所有等号情形，并得到无维稳定性结果，将原有界改进为线性界。
AI 中文摘要
设$\mathbf p=(p_1,\ldots,p_n)$与$\mathbf q=(q_1,\ldots,q_n)$属于$(0,1/2]^n$，令$μ_{\mathbf p}$、$μ_{\mathbf q}$为$2^{[n]}$上的关联测度，且满足$p_1q_1=\max_{i\in[n]}p_iq_i$。我们证明每对交叉相交族$\mathcal A,\mathcal B\subseteq2^{[n]}$都满足尖锐不等式$μ_{\mathbf p}(\mathcal A)μ_{\mathbf q}(\mathcal B)\leq p_1q_1$，这证实了Suda、Tanaka与Tokushige提出的一个猜想[Math. Program. 166 (2017) 113--130]。我们还刻画了全部等号情形：当$p_1q_1<1/4$时，等号成立当且仅当两个族均由包含同一个乘积最大化坐标的所有子集构成；在端点$p_1q_1=1/4$处，我们精确识别出额外的极值对，这类极值对由满足$p_i=q_i=1/2$的坐标上的半规模递增族诱导得到。我们进一步通过证明无维稳定性定理，解决了同一篇文献中余下的猜想：假设第一个坐标在两个测度下均具有最大概率，且满足$p_1,q_1<1/2$，若$μ_{\mathbf p}(\mathcal A)μ_{\mathbf q}(\mathcal B)\geq(1-\varepsilon)p_1q_1$成立，则存在坐标$j$，使得$\mathcal A$与$\mathcal B$在各自测度下，与包含$j$的所有子集构成的族的距离均不超过$c(p_1,q_1)\varepsilon$，该结果将猜想中的$O(\sqrt{\varepsilon})$界改进为线性界。尖锐测度定理的核心新方法是结合坐标截面归纳的对数赔率插值，稳定性结论则通过半正定估计与单坐标逼近定理推导得到。
英文摘要
Let $\mathbf p=(p_1,\ldots,p_n)$ and $\mathbf q=(q_1,\ldots,q_n)$ belong to $(0,1/2]^n$, and let $μ_{\mathbf p}$ and $μ_{\mathbf q}$ be the associated measures on $2^{[n]}$. Suppose that $p_1q_1=\max_{i\in[n]}p_iq_i$. We prove that every pair of cross-intersecting families $\mathcal A,\mathcal B\subseteq2^{[n]}$ satisfies the sharp inequality $μ_{\mathbf p}(\mathcal A)μ_{\mathbf q}(\mathcal B)\leq p_1q_1$. This confirms a conjecture of Suda, Tanaka and Tokushige [Math. Program. 166 (2017) 113--130]. We also determine all equality cases. When $p_1q_1<1/4$, equality is attained only when both families consist of all subsets containing the same product-maximizing coordinate. At the endpoint $p_1q_1=1/4$, we identify precisely the additional extremal pairs, which are induced by half-sized increasing families on the coordinates satisfying $p_i=q_i=1/2$. We further resolve the remaining conjecture from the same paper by proving a dimension-free stability theorem. Assume that the first coordinate has maximum probability under both measures and that $p_1,q_1<1/2$. If $μ_{\mathbf p}(\mathcal A)μ_{\mathbf q}(\mathcal B)\geq(1-\varepsilon)p_1q_1$, then there exists a coordinate $j$ such that both $\mathcal A$ and $\mathcal B$ are within $c(p_1,q_1)\varepsilon$, in their respective measures, of the family of all subsets containing $j$. This improves the conjectured $O(\sqrt{\varepsilon})$ bound to a linear one. The main new ingredient in the sharp measure theorem is a log-odds interpolation combined with induction on coordinate sections, while stability follows from a semidefinite estimate and a one-coordinate approximation theorem.
Comments51 pages
