Inverse problems of the Erd\H{o}s-Ko-Rado type theorems for families of vector spaces and permutations

TL;DR

This paper extends the study of extremal intersecting families from subsets to vector spaces and permutations, characterizing structures that maximize total intersection numbers and providing bounds for these quantities.

Contribution

It introduces new inverse problems for Erd ext{"o}s-Ko-Rado type theorems, characterizing extremal families of vector spaces and permutations with maximal total intersection.

Findings

Structural characterizations of extremal families for certain sizes.

Maximal total intersection numbers are achieved by specific family structures.

Derived upper bounds for total intersection numbers in vector spaces and permutations.

Abstract

Ever since the famous Erd\H{o}s-Ko-Rado theorem initiated the study of intersecting families of subsets, extremal problems regarding intersecting properties of families of various combinatorial objects have been extensively investigated. Among them, studies about families of subsets, vector spaces and permutations are of particular concerns. Recently, the authors proposed a new quantitative intersection problem for families of subsets: For $\mathcal{F}\subseteq {[n]\choose k}$, define its \emph{total intersection number} as $\mathcal{I}(\mathcal{F})=\sum_{F_1,F_2\in \mathcal{F}}|F_1\cap F_2|$. Then, what is the structure of $\mathcal{F}$ when it has the maximal total intersection number among all families in ${[n]\choose k}$ with the same family size? In \cite{KG2020}, the authors studied this problem and characterized extremal structures of families maximizing the total intersection number of given sizes. In this paper, we consider the analogues of this problem for families of vector spaces and permutations. For certain ranges of family size, we provide structural characterizations for both families of subspaces and families of permutations having maximal total intersection numbers. To some extent, these results determine the unique structure of the optimal family for some certain values of $|\mathcal{F}|$ and characterize the relation between having maximal total intersection number and being intersecting. Besides, we also show several upper bounds on the total intersection numbers for both families of subspaces and families of permutations of given sizes.