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Rainbow considerations around the Hales-Jewett theorem

Amanda Montejano

Abstract

For positive integers $t$ and $n$ let $C_t^n$ be the $n$-cube over $t$ elements, that is, the set of ordered $n$-tuples over the alphabet $\{0,\dots, t-1\}$. We address the question of whether a balanced finite coloring of $C_t^n$ guarantees the presence of a rainbow geometric or combinatorial line. For every even $t\geq 4$ and every $n$, we provide a $\left(\frac{t}{2}\right)^n$--coloring of $C_t^n$ such that all color classes have the same size, and without rainbow combinatorial or geometric lines.

Rainbow considerations around the Hales-Jewett theorem

Abstract

For positive integers and let be the -cube over elements, that is, the set of ordered -tuples over the alphabet . We address the question of whether a balanced finite coloring of guarantees the presence of a rainbow geometric or combinatorial line. For every even and every , we provide a --coloring of such that all color classes have the same size, and without rainbow combinatorial or geometric lines.
Paper Structure (4 sections, 3 theorems, 21 equations, 3 tables)

This paper contains 4 sections, 3 theorems, 21 equations, 3 tables.

Table of Contents

  1. Results
  2. Proofs
  3. Future directions to work on
  4. Acknowledgements

Key Result

Theorem 1.1

For every pair of positive integers $t$ and $k$ there exists a least positive integer $HJ(t, k)$ such that, for $n\geq HJ(k,t)$, every $k$-coloring of $C_t^n$ contains a monochromatic geometric (combinatorial) line.

Theorems & Definitions (6)

  • Theorem 1.1: Hales--Jewett, hj
  • Theorem 1.3
  • Theorem 1.5
  • Example 2.1
  • proof : Proof of theorem \ref{['thm:Q1']}
  • proof : Proof of theorem \ref{['thm:main']}