Rainbow Combinatorial Lines in Hypercubes

TL;DR

This paper investigates the anti-Hales Jewett number for hypercubes, establishing bounds and exact values for specific cases, thereby advancing understanding of rainbow combinatorial lines in high-dimensional cubes.

Contribution

It introduces new bounds for the anti-Hales Jewett number in hypercubes and provides exact values for small dimensions, extending combinatorial understanding in this area.

Findings

Established bounds for $ah(k, n)$ for general $k$ and $n$

Determined exact values for $ah(3, n)$ for small $n$

Provided bounds for $ah(3, 4)$ and exact values for lower dimensions

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 $23 < ah(3, 4) \leq 27$.