On forbidden minors for (3)

A new, surprisingly simple proof is given of the finiteness of the set of matroids minor-minimally not representable over GF(3). It is, in fact, proved that every such matroid has rank or corank at most 3. Introduction. For q a prime power we denote by C(q) the class of matroids representable over GF(q), and by 7(q) the class of matroids which are minorminimally not in ?(q). It was conjectured first by Rota [R] that 7(q) is finite for each q. This had been proved earlier by Tutte [T2] for q = 2, and has since been proved in a number of ways for q = 3 (R. Reid, unpublished circa 1972, or [S, B, Ti, K]); but for larger q the conjecture remains open. Most of the above mentioned proofs proceed by actully determining the set 7(q), but, as remarked in [Kl], nothing so precise is likely to be possible for general q. Our purpose here is to present a surprisingly simple proof of the finiteness of 7(3), specifically proving THEOREM. If M E 7(3) then either M or M* has rank at most 3. (Recall M E 7(3) X M* E 7(3).) Of course this reduces the precise determination of 7(3) to the determination of its rank 3 members, a problem easily settled by ad hoc arguments. For matroid background we refer to Welsh [W], from which our notation differs in that we use r(A) and A for the rank and closure of a set A. 1. A Lemma. To a large extent our proof parallels the easier part of the argument of [S]. The new idea, which eliminates virtually all of the hard work of that paper, is a timely application of the following simple fact. LEMMA. Let M be a connected simple matroid of rank r > 2 on a set S and let X = {x E S: M/x is disconnected}. Then (a) IXI < r2, and (b) if IXI = r 2 then there are lines 1o,... ,l-2 and an orderingXl, ,X-2 of X such that