A Composition-Based Approach to EKR Problems

TL;DR

This work introduces a composition-based framework for establishing Erdős–Ko–Rado (EKR) and strong EKR properties in structured set systems. By formalizing regular relations between base and larger families and defining EKR chains, the authors prove a Composition Lemma that transfers EKR properties from components to composites, and a G-balanced Lemma that leverages transitive group actions. The framework not only yields streamlined proofs of classical results (via generalized Katona cycle methods) but also yields new asymptotic strong EKR results for copies of fixed graphs and hypergraphs, including H-copies in complete bipartite graphs and uniform hypergraphs. Across cycles, matchings, and H-copies in graphs and hypergraphs, the approach provides broad, uniform mechanisms to derive (strong) EKR properties for large ambient structures, expanding the toolkit beyond admissible-ordering-based methods. These results have potential implications for extremal combinatorics and related counting problems where symmetry and substructure containment govern maximal intersecting subfamilies.

Abstract

Let $\mathcal{A}$ be a family of subsets of a finite set. A subfamily of $\mathcal{A}$ is said to be intersecting when any two of its members contain at least one common element. We say that $\mathcal{A}$ is an Erd{\H o}s-Ko-Rado (EKR) family if, for every element $x$ of the set, the subfamily consisting of all members of $\mathcal{A}$ that contain $x$ has the maximum cardinality among all intersecting subfamilies of $\mathcal{A}$. If these subfamilies are the only maximum intersecting subfamilies of $\mathcal{A}$, then $\mathcal{A}$ is called a strong EKR family. In this article, we introduce a compositional framework to establish the EKR and strong EKR properties in set systems when some subfamilies are known to satisfy the EKR or strong EKR properties. Our method is powerful enough to yield simpler proofs for several existing results, including those derived from Katona's cycle method (1968), Borg and Meagher's admissible ordering method (2016), related results on the family of permutations studied by Frankl and Deza (1977) and the family of perfect matchings of complete graphs of even order investigated by Meagher and Moura (2005). To demonstrate the applicability and effectiveness of our method when other existing methods have not been successful, we show that for every fixed $r$-uniform hypergraph $H$ and all sufficiently large integers $n$, the family of all subhypergraphs of the complete $r$-uniform hypergraph on $n$ vertices that are isomorphic to $H$ satisfies the strong EKR property, where two copies of $H$ are considered intersecting if they share at least one common hyperedge. Moreover, when the structural constraint $H$ is restricted to be a cycle, we establish a series of EKR results for families of cycles in the complete graph $K_n$ and the complete bipartite graph $K_{n,n}$ for a broad range of the parameter $n$.

Theorems & Definitions (36)

• Definition 1: Intersecting family, EKR and strong EKR properties
• Theorem A
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 2
• Definition 3: EKR chain and special EKR chain
• Lemma 1: Composition Lemma
• Corollary 1: Erdős-Ko-Rado Theorem EKR61