The small-order extremal 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.

Small-order extremal conjecture. For any $n\leq 2\times2^{d/2}+(r-2)$, 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

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

where $P_1,\dots,P_r$ is a partition of $[d]$.

This conjecture asserts that, for the indicated small range of $n$, an extremal $K_{r+1}$-free intersection graph comes from an $r$-partite construction based on a partition of the coordinate set. The paper presents this as the natural extremal construction following the relevant Turán-type examples; no resolution is supplied here.