2 problems

Uniqueness conjecture. For every $K\geq4$, there is only one subgraph-counting identity modulo the easily described families and the difficult identities for lower $K$.

Existence conjecture. For every $K\geq 4$, there is such an identity that holds for every graph $G$ with $n$ vertices, has one graph $J_{a,i}$ or $J'_{a,j}$ with $K$ vertices, and…