Truncations of projective geometries and uniform minors

Determine, for a prime power q and integers r 2 and m 0, the maximum integer t such that the m-th truncation T^{(m)}(PG(r+m-1,q)) contains a U_{s,t}-minor, for each positive integer s.

Background

The paper studies Tur7an-type extremal questions for matroid basis hypergraphs, particularly the maximum number of bases in matroids excluding a uniform matroid as a minor. In the subsection on matroid truncations, the authors discuss constructions based on q-representable projective geometries and their m-th truncations. These constructions yield strong lower bounds for link Tur7an densities because truncation preserves useful structural properties while reducing rank.

The proposed problem asks how large a uniform minor U_{s,t} can occur in such a truncated projective geometry. The authors note that even the special case m=0 is equivalent to determining over which fields a given uniform matroid is representable, which is itself an open problem in projective geometry and coding theory. Coarse estimates for this minor parameter could have consequences for vertex Tur7an problems on the hypercube.

References

For all $s$, what is the maximum $t$ so that $T{(m)}(PG{r+m-1}{q})$ has a $U_{s,t}$-minor?

— Turán densities for matroid basis hypergraphs  (2502.03673 - Pol et al., 5 Feb 2025) in Problem 1, Section 7.1 (Matroid truncations)

It will certainly be difficult to give an exact answer. Even when $m=0$, the problem is equivalent to the question over which fields the uniform matroid $U_{s,t}$ is representableProblem~6.5.19, an open problem in projective geometry and coding theory.

— Turán densities for matroid basis hypergraphs  (2502.03673 - Pol et al., 5 Feb 2025) in Section "Directions for future work," subsection "Matroid truncations," Problem 1 (Problem \ref{prob: projections of projective geometries})