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A note on strong Erdős-Hajnal for graphs with bounded VC-minimal complexity

Yayi Fu

TL;DR

For any $N in\mathbb{N}^{>0}$, there is $k_N>0$ such that for any finite bipartite graph $(X,Y;E) with VC-minimal complexity $<N, there exist $X'\subseteq X$ and $Y'\ subseteq Y$.

Abstract

Inspired by Adler's idea on VC minimal theories \cite{adler2008theories}, we introduce VC-minimal complexity. We show that for any $N\in\mathbb{N}^{>0}$, there is $k_N>0$ such that for any finite bipartite graph $(X,Y;E)$ with VC-minimal complexity $< N$, there exist $X'\subseteq X$, $Y'\subseteq Y$ with $|X'|\geq k_N |X|$, $|Y'|\geq k_N |Y|$ such that $X'\times Y' \subseteq E$ or $X'\times Y'\cap E=\emptyset$.

A note on strong Erdős-Hajnal for graphs with bounded VC-minimal complexity

TL;DR

For any , there is such that for any finite bipartite graph <N, there exist and .

Abstract

Inspired by Adler's idea on VC minimal theories \cite{adler2008theories}, we introduce VC-minimal complexity. We show that for any , there is such that for any finite bipartite graph with VC-minimal complexity , there exist , with , such that or .
Paper Structure (4 sections, 3 theorems, 19 equations)

This paper contains 4 sections, 3 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof
  4. Corollary

Key Result

Theorem 1.1

For $N>0$, let $k_N=\dfrac{1}{2^{N+4}}$. If a finite bipartite graph $(X,Y;E)$ has VC-minimal complexity $<N$ then there exist $X'\subseteq X$, $Y'\subseteq Y$ with $|X'|\geq k_N|X|$, $|Y'|\geq k_N|Y|$ such that $X'\times Y'\subseteq E$ or $X'\times Y'\cap E=\emptyset$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 3.1
  • proof
  • Claim 3.2
  • proof
  • ...and 7 more