Balanced supersaturation in dimensions greater than three

Prove the balanced supersaturation conjecture for the hypergraph of (d+1)-subsets of a set U⊆F_q^d that are contained in no (d−1)-flat, for every integer d>2; specifically, construct a subhypergraph H_U⊆S_{d+1}(U) satisfying |H_U|=Ω(|U|^{d+1}/q) and Δ_i(H_U)=O(|U|^{d+1−i}/q^{1−(i−1)/d}) for every i∈{1,…,d}, whenever |U|>T(d)q.

Background

For each set U⊆F_q^d, the paper defines S_{d+1}(U) as the family of (d+1)-subsets that lie in a common (d−1)-flat, equivalently the configurations that violate general position. A balanced supersaturation result seeks a large subfamily of these configurations while controlling all codegrees Δ_i.

The authors prove this conjecture for d=3, and explain that it would imply Conjecture 1 through the hypergraph container method. They explicitly identify the higher-dimensional cases as unresolved and note that new ideas are required for d>3.