Matroid minors

这一建议与1935年惠特尼在其关于矩阵理论的开创性论文中提出的组合几何中的一个基本问题有关。我们感兴趣的是将有限集合E中的点嵌入到一个空间中（更具体地说，是一个域F上的向量空间），使得E的每个子集a张成一个规定维数rk(a)的子空间。这样的嵌入称为（E, rk）在f上的表示。Whitney问哪些这样的对（E, rk）承认在f上的嵌入。他给出了这种嵌入存在的必要条件，并为满足这些条件的对（E, rk）创造了术语“矩阵”。矩阵理论已经发展成为一个非常活跃的研究领域，在编码理论、信息论、组合优化、电气工程和理论计算机科学中有着重要的应用。

This proposal is related to a fundamental problem in combinatorial geometry posed by Whitney in his seminal paper on matroid theorey in 1935. We are interested in embedding points from a finite set E into a space (more specifically, a vector space over a field F) such that each subset A of E spans a subspace of some prescribed dimension rk(A). Such an embedding is called a representation of (E, rk) over F. Whitney asked which such pairs (E, rk) admit embeddings over F. He gave necessary conditions for the existence of such embeddings, and coined the term "matroid" for the pairs (E, rk) that satisfy these conditions. Matroid theory has since grown into a highly active research area with significant applications in coding theory, information theory, combinatorial optimization, electrical engineering, and theoretical computer science.