$pi$ a subset of the $0/1$ integer points in $t$ dimensions represented by coordinates $(x_1,dots,x_t)$.

The answer to following problem is **No**.

Is there always a relationship

$$AXleq b$$

where length of $X=(X_1,X_2)$ is $theta(t^2)$ dimensions where $X_1,X_2$ are $theta(t^2)$ dimensions on the conditions

- Non-negative solutions of $X$ are in $0/1$

2.$X_1=(X_{1a},X_{1b})$ and $X_{1a}$ is of form $(underbrace{x_1,x_1,dots,x_1}_{r_1mbox{ times}},dots,underbrace{x_t,x_t,dots,x_t}_{r_tmbox{ times}})$ where $(x_1,dots,x_t)$ are the set of points in $pi$ if $X_1$ is non-negative and $X_{1b}$ is of form $(underbrace{1-x_1,1-x_1,dots,1-x_1}_{r_1’mbox{ times}},dots,underbrace{1-x_t,1-x_t,dots,1-x_t}_{r_t’mbox{ times}})$ where $(x_1,dots,x_t)$ are the set of points in $pi$ if $X_1$ is non-negative

- In 2. for every unique $X_1$ there is an unique $X_2$?

However is there a way to characterize subsets of $0/1$ cube where there is a

single$A$ which can embed any set $pi$ in the subset as alinear subsetin above sense where only the partitions $r_i,r_i’$ differ?

If we can increase $t^2$ to arbitrary polynomials and beyond (say exp or double exp) we can cover all subsets. But for now we are focusing on the smallest region.