A Brief Introduction to Matroid Theory

1. Introdu tion Many di erent topi s in mathemati s, ranging from applied subje ts su h as optimization to pure areas su h as arrangements of hyperplanes, naturally lead to matroids. The approa h we will explore is matroid theory as an abstra tion of aÆne and proje tive geometry. Therefore the rst several se tions will survey some elementary, although perhaps unfamiliar, aspe ts of geometry. We will dis uss aÆne and proje tive geometry in general, although our main interest will be in nite geometries. While aÆne geometry has been studied, in varying degrees of generality, for thousands of years, and proje tive geometry grew out of investigations into perspe tive during the Renaissan e, hie y by Girard Desargues (1591{1661), the study of nite geometries largely began in the 1930's and 1940's with the work of su h mathematiians as Marshall Hall, Jr., Ri hard H. Bru k, and Herbert Ryser. We will fo us on basi aspe ts of aÆne and proje tive geometries; this part of our survey is not meant as an introdu tion to urrent resear h in nite geometry, whi h ontinues to be a subje t of intense resear h a tivity. At its most fundamental level, geometry is on erned with su h simple notions as points, lines, planes, and their higher-dimensional ounterparts. We an onsider su h on epts even if we do not have a notion of distan e (whi h would be made pre ise by a metri ). This is exa tly what we will onsider: non-metri geometry. Thus, we will not have angles, urvature, or even the notion of \between". At rst it may seem that su h a minimalisti version of geometry would be too limited to be interesting, but this is far from the ase.