Improving generalized cap bounds in projective 3-space

Construct, for infinitely many prime powers \(q\), subsets of \(\operatorname{PG}(3,q)\) containing no four collinear points and of size at least \((1+\varepsilon)q^2\) for some fixed \(\varepsilon>0\), or otherwise resolve the associated open finite-geometry problem identified by Blokhuis.

Background

To improve the lower bound for π(H4r)\pi(H_4^r), the paper considers sets in PG⁡(3,q)\operatorname{PG}(3,q) with no four collinear points. The authors explain that a constant-factor improvement over the known lower bound m(q)≥q2+1m(q)\ge q^2+1 would require a substantial advance in finite geometry. Specifically, a set of size at least (1+ε)q2(1+\varepsilon)q^2 for infinitely many suitable values of qq would yield a plane section of size at least (1+ε−o(1))q(1+\varepsilon-o(1))q with no four collinear points and would settle an open problem attributed to Blokhuis.

References

In fact, if we had a set $ \mathcal X\subseteq \operatorname{PG}(3,q)$ proving $m(q)\ge (1+\varepsilon)q2$ for some fixed $\varepsilon>0$ and infinitely many $q$, then by averaging over the planes of $\operatorname{PG}(3,q)$ we would get a plane section of $ \mathcal X$ of size at least $(1+\varepsilon-o(1))q$, obviously still without any four points on a line. This would settle an open problem of Blokhuis for whose resolution he offered 10,000 Hungarian forints (see e.g.Section 5, Problem 2).

— Turán densities of hypergraph augmentations  (2610.11757 - Pikhurko et al., 8 Oct 2026) in Section 5, Concluding remarks