Some aspects of the critical problem for matroids
Geoff Whittle · Bulletin of the Australian Mathematical Society · 1987
The critical problem for matroids, introduced by Crapo and Rota in [2] provides a natural and unified setting for a broad range of extremal combinatorial problems.A tangential fc-block over GF(q) is a simple matroid representable over GF(.q) with critical exponent k+1 for which every loopless minor has critical exponent as most k .Such matroids are of central importance in the critical problem, (see for example [3] or [4]).The thesis is primarily concerned with a study of tangential k-blocks.The main contribution of the thesis is to develop methods by which one begins with one tangential k-block and constructs several others.Other topics (initially studied for their role in the critical problem) develop their own momentun.Brown truncations and Dilworth truncations of matroids receive attention as objects of interest in their own right.In particular sufficient conditions are given for the simple matroid associated with a quotient of a tangential k-block over GF(q) to be also a tangential k-block over GF(q).The result relies heavily on the theory of modular flats of matroids developed in [7].As a consequence it is