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arXiv · 2608.02272

On Erd\H{o}s--Ko--Rado and Hilton--Milner Theorems for Direct Products

Tian Yao·Mengyu Cao·Kaishun Wang
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Abstract

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 Erd\H{o}s--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.

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Tian Yao, Mengyu Cao, Kaishun Wang. 2026-08-03. On Erd\H{o}s--Ko--Rado and Hilton--Milner Theorems for Direct Products. https://arxiv.org/abs/2608.02272

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