On Erdős--Ko--Rado and Hilton--Milner Theorems for Direct Products

Abstract

Let $X=X_1\dot{\cup }\cdots \dot{\cup }{X}_p$ and $\mathcal{H}(\mathbf{k})=\{F\subseteq X:|F\cap X_i|=k_i\text{ for every }i\}$. We study $t$-intersecting families in one such layer and in finite unions of layers under the ordinary condition $|A\cap B|\ge t$. A generating-set method reduces every shifted extremal non-star system to at most $pt$ supporting points. This gives coordinatewise linear Erdős--Ko--Rado thresholds, improving the previous quadratic and higher-degree hypotheses, and proves that every maximum family is a full $t$-star; no positive lower bound on each layer coordinate is needed in the multilayer result. We also solve the nontrivial problem for general $t$ under an explicit polynomial large-part hypothesis, thereby resolving a problem posed by Kwan, Sudakov and Vieira. Every maximum family is a product Hilton--Milner family ${\mathcal{H}}^{\times }(T,Y)$, where $T\subseteq X$ is a $t$-set and $Y\subseteq X\setminus T$ has at least two elements. An explicit product formula and an endpoint reduction give a finite optimization and classify all equality cases, including the family ${\mathcal{A}}_1^{\times }(T\cup Y)$ when $|Y|=2$.