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Turán's theorem for Dowling geometries

Rutger Campbell, Donggyu Kim, Jorn van der Pol

TL;DR

This work extends Turán-type extremal theory to Dowling geometries Q_n(Γ) by establishing exact and asymptotic bounds for forbidding subgeometries Q_t(Γ') and M(H). The authors connect Dowling geometries to gain-graphs and their frame matroids FM(K_n^Γ), enabling reductions to subgraph conditions on gain graphs and analysis via joints and very long lines. They prove an exact extremal formula ex(Q_n(Γ), Q_t(Γ')) = |Q_n(Γ)| - n + t - 1 for nontrivial Γ' and derive a full Mantel-type theory for t=3 and partial results for t=4, highlighting substantial Γ-dependence. They also develop a general upper bound for ex(Q_n(Γ), N) across subgroups, provide π-values for graphic targets, and give detailed results for large cliques in Z_2-gain Dowling geometries, including exact and asymptotic behaviors. Overall, the paper reveals a rich interaction between group structure, gain-graphic representations, and matroidal Turán-type extremal phenomena, extending classical Turán theory from graphs to a broad matroid framework.

Abstract

The Dowling geometry $Q_n(Γ)$, where $Γ$ is a finite group, is a matroid that generalizes the complete-graphic matroid $M(K_{n+1})$. We determine the maximum size of an $N$-free submatroid of $Q_n(Γ)$ for various choices of $N$, including subgeometries $Q_m(Γ')$, lines $U_{2,\ell}$, and graphic matroids $M(H)$. When the group $Γ$ is trivial and $N=M(K_t)$, this problem reduces to Turán's classical result in extremal graph theory. We show that when $Γ$ is nontrivial, a complex dependence on $Γ$ emerges, even when $N=M(K_4)$.

Turán's theorem for Dowling geometries

TL;DR

This work extends Turán-type extremal theory to Dowling geometries Q_n(Γ) by establishing exact and asymptotic bounds for forbidding subgeometries Q_t(Γ') and M(H). The authors connect Dowling geometries to gain-graphs and their frame matroids FM(K_n^Γ), enabling reductions to subgraph conditions on gain graphs and analysis via joints and very long lines. They prove an exact extremal formula ex(Q_n(Γ), Q_t(Γ')) = |Q_n(Γ)| - n + t - 1 for nontrivial Γ' and derive a full Mantel-type theory for t=3 and partial results for t=4, highlighting substantial Γ-dependence. They also develop a general upper bound for ex(Q_n(Γ), N) across subgroups, provide π-values for graphic targets, and give detailed results for large cliques in Z_2-gain Dowling geometries, including exact and asymptotic behaviors. Overall, the paper reveals a rich interaction between group structure, gain-graphic representations, and matroidal Turán-type extremal phenomena, extending classical Turán theory from graphs to a broad matroid framework.

Abstract

The Dowling geometry , where is a finite group, is a matroid that generalizes the complete-graphic matroid . We determine the maximum size of an -free submatroid of for various choices of , including subgeometries , lines , and graphic matroids . When the group is trivial and , this problem reduces to Turán's classical result in extremal graph theory. We show that when is nontrivial, a complex dependence on emerges, even when .

Paper Structure

This paper contains 19 sections, 32 theorems, 34 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matroids
  4. Gain graphs and Dowling geometries
  5. Z2-gain-graphic representations of M(Kt)
  6. Excluding very long lines and Dowling geometries over nontrivial subgroups
  7. Joints and very long lines
  8. Dowling geometries over nontrivial subgroups: Proof of Theorem \ref{['thm intro: excluding subgeometry']}
  9. Very long lines: Proof of Theorem \ref{['thm intro: long line']}
  10. Mantel's theorem for Dowling geometries
  11. Turán density of graphic matroids
  12. General upper bound: Proof of Theorem \ref{['thm intro: gain ESS']}
  13. The non-critical pairs (Z4,K4) and (Z2xZ2,K4) : Proof of Theorem \ref{['thm intro: order 4']}
  14. Extremal numbers of large cliques in Qn(Z2)
  15. The extremal number of M(K4) in Qn(Z2)
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $n\geq t \ge 3$ and let $\Gamma'$ be a nontrivial subgroup of $\Gamma$. Then

Figures (4)

  • Figure 1: The circuits of $\mathrm{FM}(G,\mathcal{B})$ are the cycles in $\mathcal{B}$, or subgraphs of $G$ that are isomorphic to subdivisions of the three graphs above all of whose cycles are not in $\mathcal{B}$.
  • Figure 2: The three non-switching-isomorphic $\mathbb{Z}_2$-gain graphs that realize $M(K_t)$ when $t \neq 4$. The solid black edges are labelled $1$ and the dashed red edges are labelled $-1$.
  • Figure 3: The four non-switching-isomorphic $\mathbb{Z}_2$-gain graphs realizing $M(K_4)$. The solid black edges are labelled $1$ and the dashed red edges are labelled $-1$.
  • Figure 4: $K_3^{\mathbb{Z}_2}$ (leftmost) and its subgraphs realizing $U_{3,4}$ (right four illustrations).

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2: Turán's theorem for Dowling geometries
  • Theorem 1.3: Mantel's theorem for Dowling geometries
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • ...and 43 more