The characterization of maximum complete and empty subgraphs in the quadrance graph

Let $F_q$ be the finite field with $q$ elements, and let $V_q$ be the graph on $F_q\times F_q$ whose adjacency is determined by the quadrance relation used in the paper. For a subset $U\subseteq V_q$, say that $U$ spans a complete subgraph or an empty subgraph according as every pair of vertices in $U$ is adjacent or no pair is adjacent.

Maximum subgraph conjecture. If $U$ spans a complete subgraph or an empty subgraph of order $q$ in $V_q$, then $U$ is a line in $F_q\times F_q$.

The preceding bound shows that a complete or empty subgraph of this type has at most $q$ vertices, and equality is attained by the $q$ points on a line. Computations for $q=3$ and $q=7$ support the conjectured converse.