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Hilton-Milner Theorem for the $r$-independent sets in a union of cliques

Karen Gunderson, Karen Meagher, Joy Morris, Venkata Raghu Tej Pantangi

TL;DR

This work extends Hilton–Milner type results to the $r$-independent sets in the disjoint union of $n$ copies of a clique, denoted $\Gamma_{n,k}=\cup_{i=1}^n K_k$, by determining the largest intersecting families of $\mathcal{I}_{n,k}^r$ with empty overall intersection. The authors develop a robust projection and compression framework to reduce to bounded-set questions, proving an $r$-HM analogue and a cross-intersection version, with explicit extremal structures $\mathcal{H}_{n,k}^r$ and $\mathfrak{H}_{n,k}^{r},\mathfrak{M}_{n,k}^{r}$. They then apply these results to depth-two claws, resolving the Holroyd–Talbot conjecture for all values of $r$ for these graphs and generalizing previous small-$r$ outcomes to all $r$. The methods combine stability-preserving compressions, $r$-maximal cross-intersections, and careful case analyses to obtain tight bounds and complete characterizations of extremal families. The work opens avenues for Hilton–Milner type theorems in more general unions of cliques and related projection-based extremal problems in graph-intersection theory.

Abstract

We give a Hilton-Milner Theorem for the $r$-independent sets in the graph that is the union of copies of $K_k$. That is, we determine the maximum intersecting families of $r$-independent sets in this graph, subject to the condition that the sets in a family do not all share a common element. As a by-product, we also find a tight upper bound for the sum of sizes of a pair of cross intersecting families made up of the same objects. We apply our theorem to find the largest intersecting family of $r$-independent sets in a family of graphs called ``depth-two claws". This confirms the Holroyd--Talbot conjecture for depth-two claws, extending previous results on these graphs (which covered cases where $r$ was relatively small compared to the number of vertices) to all possible values of $r$.

Hilton-Milner Theorem for the $r$-independent sets in a union of cliques

TL;DR

This work extends Hilton–Milner type results to the -independent sets in the disjoint union of copies of a clique, denoted , by determining the largest intersecting families of with empty overall intersection. The authors develop a robust projection and compression framework to reduce to bounded-set questions, proving an -HM analogue and a cross-intersection version, with explicit extremal structures and . They then apply these results to depth-two claws, resolving the Holroyd–Talbot conjecture for all values of for these graphs and generalizing previous small- outcomes to all . The methods combine stability-preserving compressions, -maximal cross-intersections, and careful case analyses to obtain tight bounds and complete characterizations of extremal families. The work opens avenues for Hilton–Milner type theorems in more general unions of cliques and related projection-based extremal problems in graph-intersection theory.

Abstract

We give a Hilton-Milner Theorem for the -independent sets in the graph that is the union of copies of . That is, we determine the maximum intersecting families of -independent sets in this graph, subject to the condition that the sets in a family do not all share a common element. As a by-product, we also find a tight upper bound for the sum of sizes of a pair of cross intersecting families made up of the same objects. We apply our theorem to find the largest intersecting family of -independent sets in a family of graphs called ``depth-two claws". This confirms the Holroyd--Talbot conjecture for depth-two claws, extending previous results on these graphs (which covered cases where was relatively small compared to the number of vertices) to all possible values of .

Paper Structure

This paper contains 20 sections, 27 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Compression and projection.
  5. Intersection and cross-intersection in bounded sets
  6. Hilton-Milner analogue in bounded sets
  7. Cross-intersecting pairs in $\mathcal{I}_{n,k}^{r}$
  8. Case (a):
  9. Case (b):
  10. Case (c):
  11. Intersecting families in $\mathcal{I}_{n,k}^{r}$
  12. Case (a):
  13. Case (b):
  14. Subcase 1:
  15. Subcase 2:
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let $n \geq 2r$, and $\mathcal{B} \subseteq \binom{[n]}{r}$. If $\mathcal{B}$ is intersecting, then $|\mathcal{B}| \leq \binom{n-1}{r-1}$. Moreover, if $n>2r$, then $|\mathcal{B}| = \binom{n-1}{r-1}$ if and only if $\mathcal{B}$ is a canonically intersecting set.

Theorems & Definitions (49)

  • Theorem 2.1: Erdős-Ko-Rado (EKR) Theorem EKR1961
  • Theorem 2.2: Hilton-Milner Theorem HM
  • Theorem 2.3: Hilton--Milner HM, Simpson SIMPSON1993
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 39 more