Partition complexity measure of the boolean cube?

Given $$n$$ points $$p_1,\dots,p_n$$ in $$\{0,1\}^d$$ my goal is to find $$m$$ index sets $$\mathcal I_1,\dots,\mathcal I_m$$ on the condition that each index set is a subset of $$\{1,\dots,n\}$$ on the conditions:

1. for every pair of points $$p_i,p_j$$ for $$i,j\in\{1,\dots,n\}$$ with $$i\neq j$$ there is a $$k\in\{1,\dots,m\}$$ such that $$i,j\in\mathcal I_k$$ holds.

2. number of facets in convex hull of all the points in each of the index sets $$\mathcal I_1,\dots,\mathcal I_m$$ is $$O(n)$$ (the polytope $$\mathcal P_k$$ with vertex points indexed by with $$p_i$$ where $$i\in\mathcal I_k$$ can be defined by intersection of $$O(n)$$ half spaces at every $$k\in\{1,\dots,m\}$$).

My problem is what is the worst case $$m$$ do I need?

Does $$m=polylog(n)$$ hold if $$d=polylog(n)$$ holds?