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If $n \geq 3$ and $K$ is a field of characteristic not dividing $n$, containing a primitive $n$-th root of unity $\zeta$, then the $3n$ points of the form $(1:-\zeta^a:0)$, $(0:1:-\zeta^b)$, $(-\zeta^... • 143k 14 votes Accepted ### Geometric interpretation of the exceptional isomorphism$PSp(4,3)=PSU(4,2^2)$I believe that a finite geometric proof is given by Jean Dieudonné here: Dieudonné, Jean, Les isomorphismes exceptionnels entre les groupes classiques finis, Can. J. Math. 6, 305-315 (1954). ZBL0055.... • 10.9k 13 votes Accepted ### About the paper by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl The proof for this appeared over a series of papers. The final one was Jan Saxl, `On Finite Linear Spaces with Almost Simple Flag-Transitive Automorphism Groups' Journal of Combinatorial Theory, ... • 1,883 12 votes Accepted ### Can all lines in the euclidian plane be ordinary? The answer is yes, by an argument using the axiom of choice. There are exactly continuum many lines in the plane, and so by the axiom of choice, we may enumerate them in a well-ordered sequence of ... 12 votes ### Is the sumset or the sumset of the square set always large? Six years later it's hard to know if this is still a question of interest. For the record (since I stumbled across the question), the answer is yes and it's a consequence of a more general phenomenon. ... 11 votes Accepted ### A vertical line with many intersections with$n$non-parallel lines For the solution see the attached figure here. It is the counterexample with$n=8$. Vertical lines represent all possible vertical lines which have less that$n$intersection points with given lines. ... • 126 10 votes ### Fano plane drawings: embedding PG(2,2) into the real plane Here is a bit more symmetric version of the picture in the accepted answer: • 40.7k 10 votes ### The Sylvester-Gallai theorem over$p$-adic fields Over any field$K$of characteristic$\neq 2$and containing$i := \sqrt{-1}$, there exists in$\mathbb{P}^2(K)$a Sylvester-Gallai configuration with$12$points given in affine coordinates by$(0,0)$... • 24.9k 10 votes Accepted ### Which finite projective planes can have a symmetric incidence matrix? The key word here is "polarity". A polarity of a projective plane with point set$P$and line set$L$is a map$\pi$from$P \cup L$to itself mapping points to lines and lines to points, ... • 5,554 8 votes Accepted ### Very symmetric quadrangle in$\Bbb CP^2$That is true, since this is so for$\mathbb RP^n$- take$n+2$vertices of a "regular simplex" in it -- i.e. take the regular simplex in$S^n$and project its vertices to$\mathbb RP^n$. In ... • 28.4k 8 votes ### Incidence geometry and matrices As Will Sawin points out, this represents only a partial answer. Theorem 5 of this paper Laison, Joshua D., and Yulan Qing. "Subspace intersection graphs." Discrete Mathematics 310, no. 23 (2010): ... • 146k 8 votes Accepted ### Synthetic projective lines Building on previous work by Paul Libois, and related to work by Libois' student Jean van Buggenhaut from 1969, Francis Buekenhout considered and solved this question in "Foundations of one ... • 146 8 votes ### Why should it be hard to generalize Dvir's proof of the finite field Kakeya conjecture to the Euclidean case? I don't entirely follow your proposed approach, but let me make a few remarks which might help you think about this: (1) Clearly this can't work, since this argument purports to show that a Kakeya ... • 11.4k 6 votes ### Synthetic projective lines Following up on Matthias Wendt's comment, the language of Moufang sets is indeed a suitable axiomatic approach to (generalizations of) projective lines. Formally speaking, a Moufang set is a set$X$... • 5,554 6 votes ### Blocking sets in three dimensional finite affine spaces Here is a slightly better lower bound. If there are fewer than$2q^2$points then there is some line that hits the set at most once. Consider the$q+1$planes containing a line containing one point. ... • 27.6k 6 votes Accepted ### Is the sumset or the sumset of the square set always large? A more general result than what you want appears as Theorem 1 in http://arxiv.org/pdf/1002.2554. (A slightly weaker result had appeared before as Theorem 3.1 in http://arxiv.org/pdf/0909.5471). ... • 7,461 6 votes ### How many squares can be formed by$n$points in general position in the plane? We can get a lower bound on the order of$n \log n$. I'll describe how to arrange$4^n$points in general position to get$n 4^{n-1}$squares. The arrangement is described recursively. For the base ... • 15.7k 5 votes ### Synthetic projective lines One defining feature of$\mathbb P^1(k)$is that it provides a sharply 3-transitive permutation representation for$\operatorname{PGL}_2(k)$. I believe that the abstraction of projective line to "... 5 votes ### Synthetic projective lines You might benefit from reading Section 5.3 of John Faulkner's book The role of nonassociative algebra in projective geometry AMS 2014. The results there may have been what you had in mind by '... • 146 5 votes Accepted ### Perfect matchings in infinite regular bipartite graphs I believe this is correct (assuming$\lambda\gt0$). If$\lambda$is infinite then each connected component of$G$has$\lambda$vertices. Since the components can be handled independently, the problem ... • 9,146 5 votes ### Point-line incidence bounds over positive characteristic fields The current best Szemeredi-Trotter bound over finite fields also holds over arbitrary fields. It is in the wonderful paper "An Improved Point-Line Incidence Bound Over Arbitrary Fields" by ... • 1,032 4 votes Accepted ### For which finite projective planes can the incidence structure be written as a circulant matrix? To answer your questions: 1) A projective plane admits a circulant incidence matrix if and only if the automorphism group contains a cyclic group acting regularly on points and regularly on blocks. ... • 1,035 4 votes Accepted ### Lower bound on the distance set using incidences of points and circles The Szemerédi–Trotter bound is known to be false for circles (it is true for circles with the same radii). There is a construction that gives$N^{2/3}|C|^{2/3}\log^{1/3}N$incidences. The current best ... • 1,032 4 votes Accepted ### Does there exist a finite hyperbolic geometry in which every line contains at least 3 points, but not every line contains the same number of points? Take a 2-$(v,4,1)$design on$v$points and delete one block, along with the four points on it. In the original system each point is on exactly$(v-1)/3$blocks, so if we assume$v\ge25$the geometry ... • 11.9k 4 votes ### Applications of small Kakeya sets over finite fields I became interested in Kakeya sets because they have the interesting property that a Kakeya set in a projective plane cannot be a subset of a blocking set, and with the exception of the full plane ... • 213 4 votes ### Incidence geometry and matrices It seems that the matrix$$A = \left(\begin{array}{cccccc} 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ ... 3 votes ### Fano plane drawings: embedding PG(2,2) into the real plane I always remember this diagram that seems impossible to find online. I have no source for it (except that I remember that it is/was engraved in the sidewalk near the Weber Building on the Colorado ... • 106 3 votes Accepted ### Are two "perfectly dense" hypergraphs on$\mathbb{N}$necessarily isomorphic? There are continuum-many pairwise non-isomorphic perfectly dense hypergraphs. Below is a sketch of a proof. Given a countably infinite field$\mathbb{K}$, the projective plane$\mathbb{KP}_2$over$\...
Is the cross-ratio well-defined even when several of the four arguments coincide? No, it is not. Modeled as homogeneous coordinates over $k^2$ you get the null vector in cases where three or more ...