A construction of 3-e.c. graphs using quadrances

A graph is $n$-e.c. ($n$-existentially closed) if for every pair of subsets $A,B$ of vertex set $V$ of the graph such that $A\cap B=\mathrm{\varnothing }$ and $\mathrm{\mid }A\mathrm{\mid }+\mathrm{\mid }B\mathrm{\mid }=n$, there is a vertex $z$ not in $A\cup B$ joined to each vertex of $A$ and no vertex of $B$. Few explicit families of $n$-e.c. are known for $n>2$. In this short note, we give a new construction of 3-e.c. graphs using the notion of quadrance in the finite Euclidean space $\text{mathbbm}{Z}_p^d$.