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Extremal graph theory and point configurations in Ahlfors-David regular sets

Alex McDonald

Abstract

We study the problem of embedding bipartite graphs in Ahlfors-David regular sets of large dimension using results from extremal graph theory. Our main theorem states that any graph satisfying a power-improving bound on the extremal number can be found in the distance graph of a sufficiently high-dimensional AD-regular set. In particular, we show that AD-regular sets of dimension greater than $\frac{d+1}{2}$ must contain even cycles of all lengths if $d\geq 3$, and must contain even cycles of length at least 6 if $d=2$. This improves the best known threshold for the problem in $d\geq 4$, and yields entirely new results in $d=2,3$, under the extra assumption of AD-regularity. We also prove analogous results for large subsets of vector spaces over finite fields, which improve the best known exponent for even cycles in all dimensions.

Extremal graph theory and point configurations in Ahlfors-David regular sets

Abstract

We study the problem of embedding bipartite graphs in Ahlfors-David regular sets of large dimension using results from extremal graph theory. Our main theorem states that any graph satisfying a power-improving bound on the extremal number can be found in the distance graph of a sufficiently high-dimensional AD-regular set. In particular, we show that AD-regular sets of dimension greater than must contain even cycles of all lengths if , and must contain even cycles of length at least 6 if . This improves the best known threshold for the problem in , and yields entirely new results in , under the extra assumption of AD-regularity. We also prove analogous results for large subsets of vector spaces over finite fields, which improve the best known exponent for even cycles in all dimensions.

Paper Structure

This paper contains 8 sections, 20 theorems, 53 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Regularization and discretization of Ahlfors-David regular sets
  4. Proofs of Theorems \ref{['ADmain']} and \ref{['FFmain']}
  5. Examples and applications
  6. Proof of Corollaries \ref{['aksADcorollary']}, \ref{['cyclesADcorollary']}, \ref{['aksFFcorollary']}, and \ref{['cyclesFFcorollary']}.
  7. The hypercube graph $Q_k$
  8. Connections to VC-dimension

Key Result

Theorem 1.5

Let $E\subset \mathbb{R}^d$ be a compact, Ahlfors-David regular set with exponent $s$, and let $G$ be a graph satisfying for some $\alpha>0$. If $s>\max(\frac{d+1}{2},\frac{1}{\alpha})$, then $\Delta_G(E)$ has non-empty interior. $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: The graphs $C_4, C_6, C_8$
  • Figure 2: The approximate 4-cycle $(x_1^\varepsilon,x_2^\varepsilon,x_3^\varepsilon,x_4^\varepsilon)$ converging to a degenerate configuration $(x_1,x_2,x_3,x_2)$.
  • Figure 3: The graphs $Q_2$ and $Q_3$.
  • Figure 4: The graphs $S_1,S_2,S_3$.
  • Figure 5: $S_3$ with an extra edge (red).

Theorems & Definitions (44)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Definition 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 34 more