In 1972, McMullen has proposed the following problem:Determine the largest number \nu(d) such that any given \nu(d) points in general position in affine d-space Rd there is a projective transformation ...
The Kobon triangle problem is an unsolved problem in combinatorial geometry first stated by Kobon Fujimura. The problem asks for the largest number N(k) of nonoverlapping triangles whose sides lie on ...
Formally, the Hadwiger conjecture is: If K is any bounded convex set in the n-dimensional Euclidean space Rn, then there exists a set of 2n scalars si and a set of 2n translation vectors vi such that ...
It is of importance in the analysis of geometric algorithms to bound the number of k-sets of a planar point set, or equivalently the number of k-levels of a planar line arrangement, a problem first st ...
Erd?s Szekeres (1935) proved the following generalisation:Theorem. For any positive integer N, any sufficiently large finite set of points in the plane in general position has a subset of N points t ...
Although polyhedra and tessellations have been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics studied were: ...
Given any partial orders ≤ and ≤* on a set X, ≤* is a linear extension of ≤ exactly when (1) ≤* is a linear order and (2) for every x and y in X, if x ≤ y, then x ≤* y. It is that second proper ...
A (non-strict) partial order is a binary relation "≤" over a set P which is reflexive, antisymmetric, and transitive, i.e., which satisfies for all a, b, and c in P:a ≤ a (reflexivity);if a ≤ b and ...
The partial order formed by three elements a, b, and c with a single comparability relationship, a ≤ b, has three linear extensions, a ≤ b ≤ c, a ≤ c ≤ b, and c ≤ a ≤ b. In all three of these e ...
Singmaster (1971) showed thatN(a) = O(\log a).\,Abbot, Erd?s, and Hanson (see References) refined the estimate. The best currently known (unconditional) bound isN(a) = O\left(\frac{(\log a)(\log \log ...
Consider k runners on a circular track of unit length. At t = 0, all runners are at the same position and start to run; the runners' speeds are pairwise distinct. A runner is said to be lonely at time ...
If F is a union-closed family of sets, the family of complement sets to sets in F is closed under intersection, and an element that belongs to at least half of the sets of F belongs to at most half of ...
The symmetric group on a finite set X is the group whose elements are all bijective functions from X to X and whose group operation is that of function composition. For finite sets, "permutations" and ...
Neil Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics. The database was at first stored on punch cards. He published selections from the d ...
Magic squares were known to Chinese mathematicians as early as 650 BC, and to Arab mathematicians possibly as early as the seventh century, when the Arabs conquered northwestern parts of the Indian su ...
