Bound the dimension of configuration varieties for expanding incidence graphs

Prove that for the point-line incidence relation in the complex plane, an ε-expander bipartite graph with n vertices on each side and n^α edges, where α>1, has configuration variety of dimension at most a constant depending only on α and ε; moreover, prove that for α>4/3−1/100 and sufficiently large n, the dimension is at most 8.

Background

For a bipartite graph G encoding a point-line incidence pattern, the configuration variety V(G,Z) parametrizes realizations of that pattern over the complex numbers. Projective transformations give a baseline dimension of 8, while known constructions can have larger dimension.

The conjecture proposes that expansion prevents the dimension from growing with n and that sufficiently dense expanding configurations near the Szemerédi–Trotter threshold have no dimension beyond the projective symmetry dimension. Establishing this would constrain the possible algebraic and arithmetic structure of highly connected incidence configurations.

References

\begin{conj} \label{dimbound} Let $Z \subset C2 \times C2$ be the point-line incidence relation in $C2$ given by eqZpointline. Suppose that $G$ is an $\epsilon$-expander bipartite graph with $n$ vertices on each side and $n\alpha$ edges for $\alpha > 1$. Then $\dim V(G,Z) \le C(\alpha, \epsilon)$. Moreover, if $\alpha > \frac{4}{3} - \frac{1}{100}$, and $n > n(\epsilon)$, then $\dim V(G, Z) \le 8$. \end{conj}

— Perspectives on the unit distance problem  (2609.10791 - Guth, 9 Sep 2026) in Conjecture \ref{dimbound}, subsection “Incidence graphs and configuration varieties”

It is believed that the dimension is exactly 8, but as far as I know this has not been proven.

— Perspectives on the unit distance problem  (2609.10791 - Guth, 9 Sep 2026) in Section 6, subsection “Incidence graphs and configuration varieties”