Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An algebraic groups perspective on Erdős-Ko-Rado

Russ Woodroofe

Abstract

We give a proof of the Erdős-Ko-Rado Theorem using the Borel Fixed Point Theorem from algebraic group theory. This perspective gives a strong analogy between the Erdős-Ko-Rado Theorem and (generalizations of) the Gerstenhaber Theorem on spaces of nilpotent matrices.

An algebraic groups perspective on Erdős-Ko-Rado

Abstract

We give a proof of the Erdős-Ko-Rado Theorem using the Borel Fixed Point Theorem from algebraic group theory. This perspective gives a strong analogy between the Erdős-Ko-Rado Theorem and (generalizations of) the Gerstenhaber Theorem on spaces of nilpotent matrices.

Paper Structure

This paper contains 11 sections, 12 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Exterior algebras and intersecting sets
  4. Algebraic groups and shifted systems
  5. Exterior algebras and projective varieties
  6. Erdős--Ko--Rado for shifted set systems
  7. Proof of the main theorem
  8. Shifting and limits of algebraic group actions
  9. Generalizing combinatorial shifting via limits of matrix group actions
  10. Diagonal matrix actions, with a relationship to algebraic shifting
  11. Towards an exterior analogue of the Hilton--Milner Theorem

Key Result

Theorem 1.1

Suppose that $k\leq n/2$. If $\mathcal{A}$ is an intersecting family of $k$-element subsets of $\left[n\right]$, then $\left|\mathcal{A}\right|\leq{n-1 \choose k-1}$. If more strongly $k<n/2$, then the equality $\left|\mathcal{A}\right|={n-1 \choose k-1}$ holds only if all sets in $\mathcal{A}$ shar

Theorems & Definitions (21)

  • Theorem 1.1: Erdős--Ko--Rado Erdos/Ko/Rado:1961
  • Theorem 1.2: Gerstenhaber Gerstenhaber:1958; Serezhkin Serezhkin:1985 removed a restriction on the field
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5: Hilton--Milner Hilton/Milner:1967
  • Lemma 2.1: deRham:1954Dibag:1974
  • Theorem 2.2: Borel Fixed-Point Theorem
  • Remark 2.3
  • Proposition 2.4: see e.g. Herzog/Hibi:2011Miller/Sturmfels:2005
  • proof
  • ...and 11 more