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Rainbow Combinatorial Lines in Hypercubes

Michael Zheng

Abstract

This paper is about the rainbow dual of the Hales Jewett number, providing general bounds an anti-Hales Jewett Number for hypercubes of length k and dimension n denoted $ah(k, n).$ The best general bounds this paper provides are: $(k-1)^n < ah(k, n) \leq \frac{(k-1)^2-2}{k-1}\cdot k^{n-1}+\frac{k+1}{k-1}.$ This paper also includes proofs about the specific cases of $k = 2$ and $k = 3$, where we show that $ah(2, n) = 2$ and $2^n < ah(3, n) \leq 3^{n-1} - 2\cdot3^{n-4} + 2$ for all natural numbers n $>$ 4. For $n < 4$, we have found the exact values: $ah(3, 1) = 3$, $ah(3, 2) = 5$, and $ah(3, 3) = 11$. In the case $n = 4$, we have found that $23 < ah(3, 4) \leq 27$.

Rainbow Combinatorial Lines in Hypercubes

Abstract

This paper is about the rainbow dual of the Hales Jewett number, providing general bounds an anti-Hales Jewett Number for hypercubes of length k and dimension n denoted The best general bounds this paper provides are: This paper also includes proofs about the specific cases of and , where we show that and for all natural numbers n 4. For , we have found the exact values: , , and . In the case , we have found that .

Paper Structure

This paper contains 8 sections, 18 theorems, 3 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Bounds for the anti-Hales Jewett Number
  3. Proofs
  4. Anti-Hales Jewett Number for 2-Hypercubes
  5. Anti-Hales Jewett Number for 3-Hypercubes
  6. Future Areas of Inquiry
  7. Acknowledgements
  8. The 6 Arrangements of Patterns 1 and 2 in $[3]^4$

Key Result

Theorem 1.1

This text is paraphrased from naslund_hales-jewett_2013 For all values k, r $\in \mathbb{N}$, there exists a number HJ(k, n) such that if N $\geq$ HJ(k, n) and the points of $[k]^N$ are colored with r colors, then $[k]^N$ contains atleast one monochromatic combinatorial line.

Figures (15)

  • Figure 1: A Rainbow-Free 4-Coloring of $[3]^2$
  • Figure 2: A Rainbow-Free 10-Coloring of $[3]^3$
  • Figure 3: $S_1$
  • Figure 4: $S_2$
  • Figure 5: Diagonal Start Points
  • ...and 10 more figures

Theorems & Definitions (36)

  • Definition 1.1: Hypercube Notation
  • Definition 1.2: Combinatorial Line
  • Definition 1.3: Rainbow Free
  • Definition 1.4: anti-Hales Jewett Number
  • Definition 1.5: Minimal Coloring
  • Theorem 1.1: Hales-Jewett (1963)
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Lemma 2.4
  • ...and 26 more