Remove the dependence on the number of distances in the finite-field subdivision threshold

Establish that the conclusions of Theorem 1.4 hold with the size threshold C=O(q^{(d+1)/2}), independently of the cardinality |R| of the set of permitted nonzero distances, for every subset E\subseteq\mathbb{F}_q^d and every R-distance subdivision of a complete graph satisfying the theorem’s distance, dimension, and spanning conditions.

Background

Theorem \ref{th:ffApplicationSmalld0} proves that a sufficiently large subset E of \mathbb{F}_q^d contains every R-distance subdivision of a complete graph whose branch vertices are separated by a prescribed logarithmic distance. Its sufficient threshold is C=72|R|^{1/2}q^{(d+1)/2}, so allowing more possible edge distances increases the required size of E.

The paper notes that the dependence on the number of distances is best possible for the general jumbled-family theorem but conjectures that it is not necessary in this finite-field distance-graph application. The stated conjecture seeks the sharper threshold C=O(q^{(d+1)/2}), matching the single-distance scale and removing the factor involving |R|.