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Results tagged with incidence-geometry
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user 34180
Abstract incidence geometries like projective spaces, polar spaces, generalized polygons, as well as incidence problems in the real or complex Euclidean spaces (eg. Szemerédi–Trotter theorem).
2
votes
Reference on the Veblen-Young characterization of projective spaces
Here are some modern treatments:
Projective Geometries: From Foundations to Applications,
Albrecht Beutelspacher (1998) - see chapter 3 for the Veblen-Young
characterisation.
Points and Lines, Erne …
- 1,197
2
votes
Covering all, but $k$ points with affine subspaces
Here are some partial answers to your question.
Let $A = A_1 \times \dots \times A_n \subseteq F^n$ be a finite grid.
Alon and Furedi proved that you need at least $\sum (\# A - 1)$ hyperplanes to …
- 1,197
3
votes
Blocking sets in three dimensional finite affine spaces
Here is an improvement of the upper bound which I found in ``The polynomial method in Galois geometries'' by Simeon Ball. See page number 4.
The known constructions are somewhat crude. For exampl …
- 1,197
6
votes
1
answer
454
views
Applications of small Kakeya sets over finite fields
It was proved by Dvir that a Kakeya set in $\mathbb{F}_q^n$ has size at least $q^n/n!$, a bound which was later improved to $q^n/2^n$.
For $n = 2$ and $q$ odd the exact bound is $q(q+1)/2 + (q-1)/2$ …
- 1,197
11
votes
2
answers
781
views
Blocking sets in three dimensional finite affine spaces
What is the smallest possible size of a set of points in $\mathbb{F}_q^3$ which intersects (blocks) every line?
Clearly the union of three affine hyperplanes that intersect in a singleton, say $x = 0 …
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