Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the maximum intersecting sets of the general semilinear group of degree $2$

Roghayeh Maleki, Andriaherimanana Sarobidy Razafimahatratra

Abstract

Let $p$ be a prime and $q = p^k$. A subset $\mathcal{F} \subset \operatorname{ΓL}_{2}(q)$ is intersecting if any two semilinear transformations in $\mathcal{F}$ agree on some non-zero vector in $\mathbb{F}_q^2$. We show that any intersecting set of $\operatorname{ΓL}_{2}(q)$ is of size at most that of a stabilizer of a non-zero vector, and we characterize the intersecting sets of this size. Our proof relies on finding a subgraph which is a lexicographic product in the derangement graph of $\operatorname{ΓL}_{2}(q)$ in its action on non-zero vectors of $\mathbb{F}_q^2$. This method is also applied to give a new proof that the only maximal intersecting sets of $\operatorname{GL}_{2}(q)$ are the maximum intersecting sets.

On the maximum intersecting sets of the general semilinear group of degree $2$

Abstract

Let be a prime and . A subset is intersecting if any two semilinear transformations in agree on some non-zero vector in . We show that any intersecting set of is of size at most that of a stabilizer of a non-zero vector, and we characterize the intersecting sets of this size. Our proof relies on finding a subgraph which is a lexicographic product in the derangement graph of in its action on non-zero vectors of . This method is also applied to give a new proof that the only maximal intersecting sets of are the maximum intersecting sets.
Paper Structure (9 sections, 18 theorems, 58 equations)

This paper contains 9 sections, 18 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminary reduction
  3. Lexicographic product and the derangement graph of $\operatorname{\Gamma L}_{2}(q)$
  4. The maximum cocliques of $\operatorname{\Gamma L}_{2}(q)$ in odd characteristic
  5. The maximum cocliques of $\operatorname{\Gamma L}_{2}(q)$ In even characteristic
  6. $k$ is not a prime power
  7. $k$ is a prime power
  8. No Hilton-Milner sets in $\operatorname{GL}_{2}(q)$
  9. Future work and concluding remark

Key Result

Theorem 1.1

Let $k\leq n$ be two positive integers such that $2k\leq n$. If $\mathcal{F}$ is a collection of $k$-subsets of $[n]$ such that $A\cap B \neq \varnothing$ for all $A,B\in \mathcal{F}$, then If $n\geq 2k+1$, then equality holds in eq1 if and only if $\mathcal{F}$ is a canonical intersecting set, that is, there exists $a\in [n]$ such that

Theorems & Definitions (35)

  • Theorem 1.1: EKR
  • Theorem 1.2: Meagher-Spiga
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Lemma 2.1: Theorem 14.7.4 in godsil2016erdos
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • ...and 25 more