Given integers $m,n\geq 1$, let $W_{n,m}$ denote the family of all sequences $S_1,S_2,\cdots,S_m$ satisfying

(1) every $S_i$ is a subset of $\{1,2,\cdots,n\}$;

(2) $\mid S_i\cap S_j\mid\geq 3$ for all $1\leq i<j\leq m$.

How to calculate $\mid W_{n,m}\mid$? Is there any known formula for $\mid W_{n,m}\mid$?

Furthermore, if we denote $W_{n,m}^{+}$ to be the subset of $W_{n,m}$ as follow： $$W_{n,m}^{+}=\{(S_1,S_2,\cdots,S_m)\in W_{n,m}:\ \mid S_1\mid+\mid S_2\mid+\cdots\mid S_m\mid\equiv 0 \pmod{2}\}$$ How to calculate $\mid W_{n,m}^{+}\mid$?