Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Minimizing Monochromatic Subgraphs of $K_{n,n}$

Charles Gong

Abstract

Given any $r$-edge coloring of $K_{n,n}$, how large is the maximum (over all $r$ colors) sized monochromatic subgraph guaranteed to be? We give answers to this problem for $r \leq 8$, when $r$ is a perfect square, and when $r$ is one less than a perfect square all up to a constant additive term that depends on $r$. We give a lower bound on this quantity that holds for all $r$ and is sharp when $r$ is a perfect square up to a constant additive term that depends on $r$. Finally, we give a construction for all $r$ which provides an upper bound on this quantity up to a constant additive term that depends on $r$, and which we conjecture is also a lower bound.

Minimizing Monochromatic Subgraphs of $K_{n,n}$

Abstract

Given any -edge coloring of , how large is the maximum (over all colors) sized monochromatic subgraph guaranteed to be? We give answers to this problem for , when is a perfect square, and when is one less than a perfect square all up to a constant additive term that depends on . We give a lower bound on this quantity that holds for all and is sharp when is a perfect square up to a constant additive term that depends on . Finally, we give a construction for all which provides an upper bound on this quantity up to a constant additive term that depends on , and which we conjecture is also a lower bound.

Paper Structure

This paper contains 9 sections, 31 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Coloring Squares
  3. Computing $g(n,r)$
  4. $g^*(t^2) = \frac{2}{t}$
  5. $g^*(r)$ for Small Values of $r$
  6. $g^*(t^2-1) = \frac{1}{t}-\frac{1}{t^2}+\frac{1}{t-1}$
  7. Universal Upper Bound on $g^*(r)$
  8. Conjectures
  9. Acknowledgements

Key Result

Theorem 1

We have $g(n,r) \geq g^*(r) \cdot n$ for all $n,r \in \mathbb{Z}^{+}$.

Theorems & Definitions (66)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Definition 2
  • ...and 56 more