Eliminating the obstructive d=4 higher-dimensional configuration

Prove that in the d=4 case with j=2, the triples from a common 3-flat associated with a given pair cannot be almost all contained in a line, or otherwise establish a construction guaranteeing conditions (i) and (ii) of balanced supersaturation simultaneously; extend the result to the analogous obstruction in all higher dimensions.

Background

The proof of balanced supersaturation in dimension three divides according to the structure supplied by Claims 6 and 7. The authors state that analogous arguments handle the cases j=3 and j=4 when d=4, but the case j=2 requires adding three points from U in the same 3-flat as a given pair.

The difficulty is that almost all such triples might be contained in a line, preventing the authors from ensuring both the required number of configurations and the required codegree bounds. The authors believe this obstruction should not occur, but explicitly state that they do not know how to prove this; the same barrier is said to persist in all higher dimensions.

References

For the case j = 2, we now need to add three points from U which are in the same 3-flat with a given pair from A2. However, it may happen that almost all such triples from the same 3-flat are actually in a line, in which case we cannot guarantee (i) and (ii) at the same time. Intuitively, we think this case should not exist at all, but we do not know how to prove it.

— Maximum number of points in general position in a random subset of finite $3$-dimensional spaces  (2503.04102 - Balogh et al., 6 Mar 2025) in Section 4, Concluding remarks, p. 11

There are many open problems in this area. One significant open problem is to generalize Theorem \ref{thmgk} to higher dimensions.

— Perspectives on the unit distance problem  (2609.10791 - Guth, 9 Sep 2026) in Section 6, subsection “Low degree algebraic structure”