The Hales-Jewett numbers $c_{n,k}$ are defined, essentially, to be the largest possible size of a subset of $\mathbb Z_k^n$ free of arithmetic progressions with the difference in $\{0,1\}^n$ and satisfying the additional restriction that the support of the initial term of the progression is disjoint from the support of its difference. A significant part of the Polymath project Density Hales-Jewett and Moser Numbers consists in determining the numbers $c_{n,3}$. We know the first seven Hales-Jewett numbers (with $k=3$): $$ c_{0,3}=1,\ c_{1,3}=2,\ c_{2,3}=6,\ c_{3,3}=18,\ c_{4,3}=52,\ c_{5,3}=150,\ c_{6,3}=450; $$ see Theorem 1.4 of the aforementioned project, or OEIS sequence A156989.
What if we drop the support disjointness restriction and define, say, $\mu_n$ to be the largest possible size of a subset of $\mathbb F_3^n$ free of arithmetic progressions with the difference in $\{0,1\}^n$? I know that $$ \mu_0=1,\ \mu_1=2,\ \mu_2=6,\ \mu_3=14,\ \text{and}\ 36\le\mu_4\le 40. $$
Virtually nothing is known about the order of growth of the numbers $\mu_n$, and to develop some intuition, it would be extremely helpful to compute several more of them.