$(\alpha,\beta,d)$-Hamming Distance Graph $G_d(\alpha,\beta)$ for $\alpha,\beta\in(0,1]$ is a graph on $2^d$ vertices $v_0,\dots,v_{2^d-1}$ with edges $(v_i,v_j)\in\mathcal E(G_d)$ iff $0<\sum_{t=1}^d|i_t-j_t|<\beta d^\alpha$ where $i_t,j_t\in\{0,1\}$ at every $t\in\{1,\dots,d\}$ are binary digits of $i$ and $j$ respectively.
Take $d=\omega(\log n)$ and pick $n$ random vertices uniformly in $G_d(\alpha,\beta)$ and consider the subgraph $H$ of the $n$ random vertices with edges derived from $G_d(\alpha,\beta)$.
What is the probability that $H$ contains a Hamiltonian path?
What is the probability that $H$ contains at least $n^{\omega(1)}$ Hamiltonian paths?
Is there a $0/1$ law in the sense for given $d=\omega(\log n)$ is there a minimum $\alpha<1$ above which probability of finding at least one Hamiltonian path is $1$ and below which it is $0$?
