Generalizing the Algebra of Throws to Rank-3 Matroids

<p>The algebra of throws is a geometric construction which reveals the underlying algebraic operations of addition and multiplication in a projective plane. In Desarguesian projective planes, the algebra of throws is a well-defined, commutative and associative binary operation. However, when we consider an analogous operation in a more general point-line configuration that comes from rank-3 matroids, none of these properties are guaranteed. We construct lists of forbidden configurations which give polynomial time checks for certain properties. Using these forbidden configurations, we can check whether a configuration has a group structure under this analogous operation. We look at the properties of configurations with such a group structure, and discuss their connection to the jointless Dowling geometries.</p>

Generalizing the Algebra of Throws to Rank-3 Matroids

<p>The algebra of throws is a geometric construction which reveals the underlying algebraic operations of addition and multiplication in a projective plane. In Desarguesian projective planes, the algebra of throws is a well-defined, commutative and associative binary operation. However, when we consider an analogous operation in a more general point-line configuration that comes from rank-3 matroids, none of these properties are guaranteed. We construct lists of forbidden configurations which give polynomial time checks for certain properties. Using these forbidden configurations, we can check whether a configuration has a group structure under this analogous operation. We look at the properties of configurations with such a group structure, and discuss their connection to the jointless Dowling geometries.</p>

Exponential multivalued forbidden configurations

The forbidden number $\mathrm{forb}(m,F)$, which denotes the maximum number of unique columns in an $m$-rowed $(0,1)$-matrix with no submatrix that is a row and column permutation of $F$, has been widely studied in extremal set theory. Recently, this function was extended to $r$-matrices, whose entries lie in $\{0,1,\dots,r-1\}$. The combinatorics of the generalized forbidden number is less well-studied. In this paper, we provide exact bounds for many $(0,1)$-matrices $F$, including all $2$-rowed matrices when $r > 3$. We also prove a stability result for the $2\times 2$ identity matrix. Along the way, we expose some interesting qualitative differences between the cases $r=2$, $r = 3$, and $r > 3$. Comment: 12 pages; v3: formatted for DMTCS; v2: Corollary 3.2 added, typos fixed, some proofs clarified

Review of Forbidden Configurations in Discrete Geometry by David Eppstein

In 1930, the mathematician Esther Klein observed that any five points in the plane in general position (i.e., no three points forming a line) contain four points forming a convex quadrilateral. This innocentsounding discovery led to major lines of research in discrete geometry. Klein's friends Paul Erdős and George Szekeres generalized this theorem, and also conjectured that 2k-2 + 1 points (again in general position) would be enough to force a convex k-gon to exist. The resolution of this conjecture became known as the "happy ending problem," because Klein and Szekeres ended up getting married. The unhappy side is that it has, to date, not been completely solved, although a recent breakthrough of Suk made significant progress. This both mathematically and personally charming little story is a great beginning for this elegant book about discrete geometry. It typifies the type of problems that are studied throughout, and also captures the spirit of curiosity that drives such studies. The book covers many problems that lie at the intersection of three fields: discrete geometry, algorithms and computational complexity.

Points and Lines

In Forbidden Configurations in Discrete Geometry, David Eppstein examines a number of seemingly simple problems in plain geometry concerning finite sets of points and lines in the plane. Although most of these problems should prove relatively easy to grasp for the uninitiated, general solutions to these problems are not known. The book will serve both budding mathematicians and curious amateurs.