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Bipartite and Euclidean Gallai-Ramsey Theory

Isabel McGuigan, Katherine Pan

Abstract

In this paper, we investigate the following Gallai-Ramsey question: how large must a complete bipartite graph $K_{n_1, n_2}$ be before any coloring of its edges with $r$ colors contains either a monochromatic copy of $G = K_{s,t}$ or a rainbow copy of $H = K_{s,t}$? We demonstrate that the answer is linear in $r$, and provide more precise bounds for the specific case $s = 2$. Furthermore, we also consider the following Euclidean Gallai-Ramsey question: given a configuration $H$ in Euclidean space, what is the smallest $n$ such that any $r$-coloring of $n$-dimensional Euclidean space contains a monochromatic or rainbow configuration congruent to $H$? Through a natural translation between edge colorings of the complete bipartite graph $K_{n_1,n_2}$ and colorings of a subset of $(n_1+n_2)$-dimensional Euclidean space, we prove new upper bounds on $n$ for some configurations which can be expressed as Cartesian products of simplices.

Bipartite and Euclidean Gallai-Ramsey Theory

Abstract

In this paper, we investigate the following Gallai-Ramsey question: how large must a complete bipartite graph be before any coloring of its edges with colors contains either a monochromatic copy of or a rainbow copy of ? We demonstrate that the answer is linear in , and provide more precise bounds for the specific case . Furthermore, we also consider the following Euclidean Gallai-Ramsey question: given a configuration in Euclidean space, what is the smallest such that any -coloring of -dimensional Euclidean space contains a monochromatic or rainbow configuration congruent to ? Through a natural translation between edge colorings of the complete bipartite graph and colorings of a subset of -dimensional Euclidean space, we prove new upper bounds on for some configurations which can be expressed as Cartesian products of simplices.

Paper Structure

This paper contains 11 sections, 13 theorems, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Gallai-Ramsey Theory
  3. Euclidean Ramsey Theory
  4. Euclidean Gallai-Ramsey Theory
  5. Main results
  6. Bipartite Gallai-Ramsey Numbers
  7. Proof of Theorem \ref{['thm:K_st']}
  8. Proof of Theorems \ref{['thm:star']} and \ref{['thm:K_2t']}
  9. Euclidean Gallai-Ramsey Numbers
  10. Future Work
  11. Acknowledgements

Key Result

Theorem 1.1

For any pair of positive integers $s, t$, there exists a constant $C_{s,t}$ such that, for $n = C_{s,t}r$, every $r$-coloring of $K_{n,n}$ contains a monochromatic or rainbow $K_{s,t}$.

Figures (3)

  • Figure 1: Visualization of the paths $\pi_k$
  • Figure 2: All the paths $\pi_k$ adjacent to $u_i$ and $u_j$: at most $(t-1)r$ of them are monochromatic, and at most $t-1$ of them belong to $\mathcal{P}$.
  • Figure 3: A visualization of $\phi$ from the edges of $K_{2,3}$ to an equilateral triangular prism (the Cartesian product of a regular 1-simplex with side length $a$ and a regular 2-simplex with side length $b$). For example, $\phi$ maps $(v_1,u_1)$ to $x_{1,1} = (\frac{a}{\sqrt{2}}, 0, \frac{b}{\sqrt{2}},0,0)$. Note that the figure on the right is a 3-dimensional projection of the resulting prism, which lies in 5-dimensional space.

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2: also proven in eroh04
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1: bollobas
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 14 more