The large-order extremal construction conjecture for subcube intersection graphs

Let $C_d$ be the family of subcubes of $\{0,1\}^d$, and for a subcube $x\in C_d$ let $F(x)\subseteq[d]$ denote its fixed-coordinate set. Let $\mathcal{I}(n,d)$ be the family of intersection graphs of $n$ subcubes in $C_d$, and let $K_{r+1}$ denote the complete graph on $r+1$ vertices.

Large-order extremal construction conjecture. For any $n\geq 2\times2^{d/2}+(r-2)$, the maximum number of edges in a $K_{r+1}$-free graph in $\mathcal{I}(n,d)$ is attained by a graph $G$ which is a subgraph of a graph $H$ with

$$V(H)=\{x\in C_d:F(x)=R_i\},$$

where, if $P_i=[d]\setminus R_i$, then $P_1,\dots,P_k$ is a partition of $[d]$ for some $k$, and $P_{k+1},\dots,P_r=[d]$.

This complements the proposed small-order construction by allowing fewer than $r$ nontrivial classes and additional classes based on the full fixed-coordinate set. The paper motivates the construction through lower bounds for large $n$ but does not establish its optimality.