= Solution
For an unrestricted binary relation $R$, the extension axioms permit arbitrary directed edges and loops. For each $m\ge0$ and each choice of bits $\epsilon_i,\delta_i,\eta\in\{0,1\}$, require that every tuple of distinct $x_1,\ldots,x_m$ has a fresh element $y$ with
$$
y\ne x_i\quad\text{for all }i,\qquad
R(y,y)^\eta\land\bigwedge_{i=1}^m
\bigl(R(y,x_i)^{\epsilon_i}\land R(x_i,y)^{\delta_i}\bigr).
$$
Here $P^1$ denotes $P$ and $P^0$ denotes $\neg P$; the notation is a literal abbreviation, not exponentiation of a relation. Incoming and outgoing choices are independent, and the loop bit is also specified. These are the <extension axioms for an unrestricted binary relation>. If one instead restricts to undirected loop-free graphs, the allowable patterns reduce to the random-graph extension axioms of Question 2; no such restriction is imposed here.
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