Editing Matroid (section)

==History==
Matroid theory was introduced by {{harvp|Whitney|1935}}. It was also independently discovered by [[Takeo Nakasawa]], whose work was forgotten for many years ({{harvp|Nishimura|Kuroda|2009}}).

In his seminal paper, Whitney provided two axioms for independence, and defined any structure adhering to these axioms to be "matroids".{{efn|
Although it was perhaps implied, {{harvp|Whitney|1935}} did not include an axiom requiring at least one subset to be independent.
:
}}
His key observation was that these axioms provide an abstraction of "independence" that is common to both graphs and matrices.
Because of this, many of the terms used in matroid theory resemble the terms for their analogous concepts in [[linear algebra]] or [[graph theory]].

Almost immediately after Whitney first wrote about matroids, an important article was written by {{harvp|MacLane|1936}} on the relation of matroids to [[projective geometry]]. A year later, {{harvp|van der Waerden|1937}} noted similarities between algebraic and linear dependence in his classic textbook on Modern Algebra.

In the 1940s [[Richard Rado]] developed further theory under the name "independence systems" with an eye towards [[Transversal (combinatorics)|transversal theory]], where his name for the subject is still sometimes used.

In the 1950s [[W. T. Tutte|W.T. Tutte]] became the foremost figure in matroid theory, a position he retained for many years. His contributions were plentiful, including
* the characterization of [[binary matroid|binary]], [[regular matroid|regular]], and [[graphic matroid|graphic]] matroids by [[Matroid minor|excluded minors]]

* the regular-matroid representability theorem

* the theory of chain groups and their matroids

and the tools he used to prove many of his results:

* the "Path theorem"

* "[[Tutte homotopy theorem]]" (see, e.g., {{harvp|Tutte|1965}})

which are so complicated that later theorists have gone to great trouble to eliminate the need for them in proofs.{{efn|
A fine example is [[A. M. H. Gerards|A.M.H. Gerards']] short proof ({{harvp|Gerards|1989}}) of Tutte's characterization of regular matroids.
}}

{{harvp|Crapo|1969}} and {{harvp|Brylawski|1972}} generalized to matroids Tutte's "dichromate", a graphic polynomial now known as the [[Tutte polynomial]] (named by Crapo).  Their work has recently (especially in the 2000s) been followed by a flood of papers&mdash;though not as many as on the Tutte polynomial of a graph.

In 1976 [[Dominic Welsh]] published the first comprehensive book on matroid theory.

[[Paul Seymour (mathematician)|Paul Seymour]]'s decomposition theorem for regular matroids ({{harvp|Seymour|1980}}) was the most significant and influential work of the late 1970s and the 1980s.
Another fundamental contribution, by {{harvp|Kahn|Kung|1982}}, showed why projective geometries and [[Dowling geometry|Dowling geometries]] play such an important role in matroid theory.

By the 1980s there were many other important contributors, but one should not omit to mention [[Geoff Whittle]]'s extension to ternary matroids of Tutte's characterization of binary matroids that are representable over the rationals {{harv|Whittle|1995}}, perhaps the biggest single contribution of the 1990s.

In the current period (since around 2000) the Matroid Minors Project of [[Jim Geelen|Geelen]], Gerards, Whittle, and others,{{efn|
The Matroid Minors Project is an attempt to duplicate, for matroids that are representable over a finite field, the success of the Robertson–Seymour Graph Minors Project (see [[Robertson–Seymour theorem]]).
}}
has produced substantial advances in the structure theory of matroids. Many others have also contributed to that part of matroid theory, which (in the first and second decades of the 21st&nbsp;century) is flourishing.