mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Vámos matroid or Vámos cube is a

matroid In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being ...

over a set of eight elements that cannot be represented as a matrix over any field. It is named after English mathematician Peter Vámos, who first described it in an unpublished manuscript in 1968.

Definition

The Vámos matroid has eight elements, which may be thought of as the eight vertices of a

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

cuboid In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a c ...

. The matroid has rank 4: all sets of three or fewer elements are independent, and 65 of the 70 possible sets of four elements are also independent. The five exceptions are four-element circuits in the matroid. Four of these five circuits are formed by faces of the cuboid (omitting two opposite faces). The fifth circuit connects two opposite edges of the cuboid, each of which is shared by two of the chosen four faces. Another way of describing the same structure is that it has two elements for each vertex of the

diamond graph In the mathematical field of graph theory, the diamond graph is a planar, undirected graph with 4 vertices and 5 edges. It consists of a complete graph minus one edge. The diamond graph has radius 1, diameter 2, girth 3, chro ...

, and a four-element circuit for each edge of the diamond graph.

Properties

*The Vámos matroid is a

paving matroid In the mathematical theory of matroids, a paving matroid is a matroid in which every circuit has size at least as large as the matroid's rank. In a matroid of rank r every circuit has size at most r+1, so it is equivalent to define paving matroids ...

, meaning that all of its circuits have size at least equal to its

rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...

.. *The Vámos matroid is isomorphic to its

dual matroid In matroid theory, the dual of a matroid M is another matroid M^\ast that has the same elements as M, and in which a set is independent if and only if M has a basis set disjoint from it... Matroid duals go back to the original paper by Hassler ...

, but it is not identically self-dual (the isomorphism requires a nontrivial permutation of the matroid elements). *The Vámos matroid cannot be represented over any field. That is, it is not possible to find a

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

, and a system of eight vectors within that space, such that the matroid of

linear independence In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...

of these vectors is isomorphic to the Vámos matroid. Indeed, it is one of the smallest non-representable matroids, and served as a counterexample to a conjecture of Ingleton that the matroids on eight or fewer elements were all representable. *The Vámos matroid is a

forbidden minor In graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to the family and further exclude all graphs from the family which contain any of these forbidde ...

for the matroids representable over a field

F

, whenever

F

has five or more elements., p. 511. However, it is not possible to test in

polynomial time In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...

whether it is a minor of a given matroid

M

, given access to

M

through an independence oracle. *The Vámos matroid is not algebraic. That is, there does not exist a

field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...

L/K

, and a set of eight elements of

L

, such that the

transcendence degree In abstract algebra, the transcendence degree of a field extension ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ...

of sets of these eight elements equals the rank function of the matroid. *The Vámos matroid is not a secret-sharing matroid. Secret-sharing matroids describe "ideal"

secret sharing Secret sharing (also called secret splitting) refers to methods for distributing a secret among a group, in such a way that no individual holds any intelligible information about the secret, but when a sufficient number of individuals combine t ...

schemes in which any coalition of users who can gain any information about a secret key can learn the whole key (these coalitions are the dependent sets of the matroid), and in which the shared information contains no more information than is needed to represent the key. These matroids also have applications in

coding theory Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studied ...

. *The Vámos matroid has no adjoint. This means that the

dual lattice In the theory of lattices, the dual lattice is a construction analogous to that of a dual vector space. In certain respects, the geometry of the dual lattice of a lattice L is the reciprocal of the geometry of L , a perspective which underlie ...

of the

geometric lattice In the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumption of finiteness. Geometric lattices and matroid lattices, r ...

of the Vámos matroid cannot be order-embedded into another geometric lattice of the same rank. *The Vámos matroid can be

oriented In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...

. In oriented matroids, a form of the

Hahn–Banach theorem The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...

follows from a certain intersection property of the flats of the matroid; the Vámos matroid provides an example of a matroid in which the intersection property is not true, but the Hahn–Banach theorem nevertheless holds. *The

Tutte polynomial The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays an important role in graph theory. It is defined for every undirected graph G and contai ...

of the Vámos matroid is. *:

x^4+4x^3+10x^2+15x+5xy+15y+10y^2+4y^3+y^4.

References

{{DEFAULTSORT:Vamos matroid Matroid theory