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A Polynomial Ramsey Statement for Bounded VC-dimension

Tomáš Hons

Abstract

A theorem by Ding, Oporowski, Oxley, and Vertigan states that every sufficiently large bipartite graph without twins contains a matching, co-matching, or half-graph of any given size as an induced subgraph. We prove that this Ramsey statement has polynomial dependency assuming bounded VC-dimension of the initial graph, using the recent verification of the Erdős-Hajnal property for graphs of bounded VC-dimension. Since the theorem of Ding et al. plays a role in (finite) model theory, which studies even more restricted structures, we also comment on further refinements of the theorem within this context.

A Polynomial Ramsey Statement for Bounded VC-dimension

Abstract

A theorem by Ding, Oporowski, Oxley, and Vertigan states that every sufficiently large bipartite graph without twins contains a matching, co-matching, or half-graph of any given size as an induced subgraph. We prove that this Ramsey statement has polynomial dependency assuming bounded VC-dimension of the initial graph, using the recent verification of the Erdős-Hajnal property for graphs of bounded VC-dimension. Since the theorem of Ding et al. plays a role in (finite) model theory, which studies even more restricted structures, we also comment on further refinements of the theorem within this context.

Paper Structure

This paper contains 11 sections, 10 theorems, 10 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. VC-dimension
  4. Growth function
  5. Erdős-Hajnal property
  6. Proof
  7. Switch submatrix
  8. Upper graph and its VC-dimension
  9. Homogenization
  10. Remarks on tame classes
  11. Acknowledgement

Key Result

Theorem 1

There is a monotone unbounded function $f: \mathbb{N} \to \mathbb{N}$ such that the following holds. Let $G = (U,V,E)$ be a bipartite graph such that the part $V$ has $n$ vertices and no twins. Then it contains an induced subgraph $F=(U',V',E')$ with $|U'| = |V'| \geq f(n)$ that is isomorphic to eit

Figures (1)

  • Figure 1: A diagram of various tameness notions and their inclusions.

Theorems & Definitions (22)

  • Theorem 1: Ding_1996Alekseev1997Gravier2004
  • Theorem 2
  • Theorem 3
  • Definition 1: VC-dimension of a set system
  • Definition 2: VC-dimension of a graph
  • Definition 3: VC-dimension of a matrix
  • Definition 4: Growth function of a set system
  • Theorem 4: Sauer-Shelah lemma Sauer_1972Shelah_1972
  • Theorem 5: NSS_2023
  • Definition 5
  • ...and 12 more