The exact density of E(d,i)

Let $d$ and $i$ be positive integers with $1\leq i<d$. Define

$$E(d,i)=\left\{(x_1,x_2,\ldots,x_d)\in V_d \mathrel{\bigg|} \sum_{j=1}^i x_j \text{ and }\sum_{j=i+1}^d x_j\text{ are both even}\right\}.$$

Let $i(K_{i,d-i})$ denote the inducibility of the complete bipartite graph $K_{i,d-i}$. E(d,i) density conjecture. Equality holds in the lower bound

$$\lambda(E(d,i),d)\geq i(K_{i,d-i}).$$

The lower bound is obtained from a bipartite blow-up construction, with the optimizing parameter given by the corresponding complete-bipartite inducibility problem. Equality is known for $E(4,1)$ and $E(4,2)$, while the conjecture asserts equality for all admissible $d$ and $i$.