AI 中文总结
该论文研究直积的Erdős–Ko–Rado与Hilton–Milner定理，通过生成集方法得到线性阈值，解决Kwan等人的问题，分类出所有等号情形的最大族。
AI 中文摘要
设$X=X_1\boldsymbol{\u0307}\u22c5\boldsymbol{\u0307}\u22c5\boldsymbol{\u0307}\u222a\boldsymbol{\u0307}\u22c5\boldsymbol{\u0307}\u22c5\boldsymbol{\u222a}\u22ef\boldsymbol{\u0307}\u22c5\boldsymbol{\u0307}\u22c5\boldsymbol{\u222a}X_p$，且$\boldsymbol{\u039b}(k)=\boldsymbol{\u007b}F\boldsymbol{\u2286}X:|F\boldsymbol{\u2229}X_i|=k_i\text{ 对每个 }i\boldsymbol{\u007d}$。我们研究这类层及有限层并集中的$t$-相交族，满足普通条件$|A\boldsymbol{\u2229}B|\boldsymbol{\u2265}t$。生成集方法将每个移位的极值非星系统简化为至多$pt$个支撑点，得到坐标意义下线性的Erdős–Ko–Rado阈值，改进了此前的二次及更高次假设，证明每个最大族是完整的$t$-星；多层结果中无需各层坐标的正下界。我们还在显式多项式大分量假设下解决了一般$t$的非平凡问题，从而解决了Kwan、Sudakov和Vieira提出的问题。每个最大族是乘积Hilton–Milner族$\boldsymbol{\u039b}^\boldsymbol{\u00d7}(T,Y)$，其中$T\boldsymbol{\u2286}X$是$t$-集，$Y\boldsymbol{\u2286}X\backslash T$至少含两个元素。显式乘积公式与端点约简给出有限优化，并对所有等号情形分类，包括$|Y|=2$时的族$\boldsymbol{\u039b}_1^\boldsymbol{\u00d7}(T\boldsymbol{\u222a}Y)$。
英文摘要
We investigate $t$-intersecting families in direct-product set systems obtained by prescribing the number of selected elements in each part of a partitioned ground set. For a finite union of layers, we prove an Erdős--Ko--Rado result under coordinatewise linear part-size conditions. As an application, we establish a new range of parameters for which a conjecture of Frankl et al.\ [\emph{J. Combin. Theory Ser. A} \textbf{155} (2018), 493--502] holds. Under an explicit polynomial large-part hypothesis, we also characterize the maximum nontrivial $t$-intersecting families for the single-layer setting. In particular, for $t=1$, this answers the problem of Kwan et al.\ [\emph{J. Combin. Theory Ser. A} \textbf{156} (2018), 44--60] asking for a classification of all extremal families, including the possible non-shifted maximizers.
