The Induced Bipartite Ramsey Theorem: An Exposition

William Gasarch, Gary Peng

Abstract

We present an exposition of the proof of the induced bipartite Ramsey Theorem.

The Induced Bipartite Ramsey Theorem: An Exposition

Abstract

We present an exposition of the proof of the induced bipartite Ramsey Theorem.

Paper Structure (9 sections, 7 theorems, 8 equations, 15 figures)

This paper contains 9 sections, 7 theorems, 8 equations, 15 figures.

Table of Contents

Introduction
Bipartite Graphs
Preliminaries
Hypergraphs
The Bipartite Ramsey Theorem
The Induced $B_{4,2}$ Ramsey Theorem
The Induced $B_{n,k}$ Ramsey Theorem
The Induced Bipartite Ramsey Theorem For a Particular Graph
Acknowledgements

Key Result

Theorem 3.3

Let $a,c,k\in\sf N$. Then there exists $n$ such that, for all $c$-colorings of $\binom{[n]}{a}$, there exists a homogenous set of size $k$.

Figures (15)

Figure 2.1: A Bipartite Graph
Figure 2.2: $K_{3, 3}$
Figure 2.3: $B_{4, 2}$
Figure 2.4: The red subgraph is not an induced subgraph.
Figure 2.5: The blue subgraph is an induced subgraph.
...and 10 more figures

Theorems & Definitions (17)

Definition 2.3
Definition 2.4
Definition 3.1
Definition 3.2
Theorem 3.3
Theorem 3.5
proof
Theorem 4.1
proof
Theorem 5.1
...and 7 more