Rainbow Combinatorial Lines in Hypercubes

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 $ah(k,n)\mathrm{.}$ The best general bounds this paper provides are: $(k-1{)}^n<ah(k,n)\le \frac{(k-1{)}^2-2}{k-1}\cdot k^{n-1}+\frac{k+1}{k-1}\mathrm{.}$ This paper also includes proofs about the specific cases of $k=2$ and $k=3$, where we show that $ah(2,n)=2$ and $2^n<ah(3,n)\le 3^{n-1}-2\cdot 3^{n-4}+2$ 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 $23<ah(3,4)\le 27$.