 Solution

Use the Rado graph on the natural numbers. Interpret the binary relation byThe relation is symmetric by its definition, and $E(i,i)$ is always false. Thus this first-order structure satisfies $F_1$ and $F_2$.

To verify an extension axiom, take distinct $x_1,\dots ,x_n$. Choose a natural number $k$ larger than every $x_i$ and putThe powers are distinct, so there are no carries in this binary representation. In particular $y>x_i$ for every $i$, and the bit in position $x_i$ is one exactly when $i<m$. Therefore $y$ is different from all the given vertices, is adjacent to $x_1,\dots ,x_{m-1}$, and is not adjacent to the others. If the $x_i$ are not distinct, the antecedent of the extension axiom is false, so it places no restriction. The case $n=0$, if included in the convention for natural numbers, simply asks for a nonempty domain, which this first-order structure has.

For example, prescribing adjacency to $1$ and $3$ and nonadjacency to $2$ permits $k=4$ and $y=16+2+8=26$, whose binary representation is $11010$. This explicitly realizes the required pattern. HenceReordering a tuple makes the extension axioms applicable to any two disjoint finite sets of vertices, with any desired adjacency and nonadjacency pattern. This is the extension property defining the theory of the random graph.