Mathematical Sciences: Growth Rates of Minor-closed Classes of Matroids

本课题主要研究拟阵理论中的两个问题。首先证明了拟阵的次闭类的尺度函数是无穷的、指数的或多项式的。其次证明了次闭类C的尺度函数有界于线性函数当且仅当C中拟阵的临界指数有一个有限的上界。拟阵理论是图论和射影几何的自然推广。因此，它在组合论中占有中心地位。

This project focusses on two problems in matroid theory. The first is to show that the size functions of minor-closed classes of matroids are either infinite, exponential, or polynomial. The second is to show that the size function of a minor-closed class C is bounded by a linear function if and only if there is a finite upper bound on the critical exponent of matroids in C. Matroid theory is a natural generalization of graph theory and projective geometry, and thus ocupies a central place in combinatorial theory.