bases in infinite matroids

BASES IN INFINITE MATROIDS RON AHARONI AND MAURICE POUZET

ABSTRACT We consider bases in matroids of infinite rank, and prove: (a) the existence of a perfect matching in the 'transition graph' of any two bases. This is an extension of the existence of a non-zero generalized diagonal in the transition matrix between bases in finite dimensional linear spaces, and settles a conjecture of the second author [8]. (b) A Cantor-Bernstein theorem for matroids. (c) The existence of a winning strategy for the 'good guy' in an exchange game between bases in infinite matroids.

1. Preliminaries A matroid is a pair Jt = (S,«/), where J is a non-empty family of subsets of S satisfying: (a) /e |/| then / u {x}eJ for some xeJ\I, and (c) if allfinitesubsets of/belong t o , / then IEJ (that is, M is offinite character). When S is infinite such structures are also called independence spaces [12]. By (c) and Zorn's lemma, J has elements which are maximal with respect to containment, and these are called bases for M. The subsets of S which belong to ./ are called independent and those which do not belong to J are said to be dependent. A minimal dependent set is called a circuit. The following is well known (see, for example, [12]). LEMMA 1.1. If B is a basis and x^B then there exists a unique subset D of B such that D U {x} is a circuit.

For B, x, D as in the lemma we write D = sB(x) (the s stands for ' support'—think of the case of linear spaces). We extend this definition by writing sB(x) = {x} for x e B. For a subset T of S we let sp (T) = T U {x e S: IU {*} i J for some independent subset / of T). If sp(r) = S we say that T is spanning (for Ji). The following lemma, which will be basic in our arguments, follows easily from axioms (a), (b) and (c). LEMMA 1.2. Ifle J, X is afinitesubset ofI,Y^S is not spanning.

and \ Y\ < \X\ then (I\X) U Y

It is not too difficult to prove the lemma also for X infinite, which yields as a corollary that all bases have the same cardinality. Since this version of the lemma will not be used, we leave its proof to the interested reader. Received 2 May 1989. 1991 Mathematics Subject Classification 05B35. Research supported by the CNRS and the PRC Math-Info. J. London Math. Soc. (2) 44 (1991) 385-392 13

JLM 44

386

RON AHARONI AND MAURICE POUZET

Let us also recall the well known 'exchange' property: if B and C are bases of a matroid then for every bsB there exists CEC such that {B\{b}) U {c} is a basis. (We shall abbreviate the notation for this set as B-b + c.) We shall also need some graph theoretic terminology. All graphs considered will be undirected, unless the contrary is explicitly stated. If G = {V, E) is a graph, F^ E, xeV and A £ V, we write

F(x} = {yeV: (x, y) e F) and F[A) = (J {F(x} :xeA}. A subset F of £ is a matching if its elements are disjoint edges. A matching F is called perfect if it covers all vertices, that is, F[V] = V. If F is a matching and xeK we write F(x) = y if (.x, J ^ E F . A subset Zof Kis called matchable if there exists a matching Fsuch that F[V] 3 X. A bipartite graph is a triple (5, C, £), where B and C are disjoint, B U C is the vertex set of the graph, E its edge set, and all elements of E are the form (b, c), beB, ceC. It is called espousable if B is matchable. If B is a base of a matroid Jl and /is an independent set in M we form a bipartite

graph (B,I,E), where £ is defined by E(i) = sB{i) for every iel. If /(15 # 0 we should in fact, first make the two sides of the graph disjoint, which could be obtained for example by taking the sides to be B x {0} and Ix {1}. Instead we just keep in mind that if a vertex belongs to B n /, then its two copies, in B and /, are considered distinct. This graph is called the B-I transition graph. (If B and / are bases in a finitedimensional vector space then the graph is the adjacency graph of the transition matrix between B and /, hence the name.) Let F = (B, C, E) be a bipartite graph. A subset D of B is called critical if it is matchable, but F[D] = E[D] for every matching F of D (that is, matching Fsuch that F[C] = D). A \-obstruction in F is a pair (a,D) such that D is a critical subset of B, asB\D, and E(a) £ E[D]. Clearly, if T contains a 1-obstruction it is inespousable. Podewski and Steffens proved [7] that for countable bipartite graphs the converse also holds, that is, if T is countable and does not possess a 1-obstruction then it is espousable. Nash-Williams introduced [3] another criterion, which he named '^-admissibility', and proved the following. THEOREM 1.3 [6]. Let T = (B, C, E) and assume that E(c} is countable for every ceC. Then T is espousable if and only if it is q-admissible.

We shall not define here ^-admissibility, since the following result makes it unnecessary for us to do so. 1.4 [1]. A bipartite graph is q-admissible if and only if it does not contain a \-obstruction. THEOREM

If N is a matching then an ^-alternating path is a finite or infinite sequence of the for form xo,y1,x1,y2,x2,... where all xt and^ are distinct,^ = N(xt) andyteE^x^} all i ^ l . The following lemma is easy. LEMMA 1.5 [7]. If D is critical and N is a matching of D then there does not exist an infinite N-altemating path.

BASES IN INFINITE MATROIDS

387

(The reason is that if such a path xo,y1,x1,y2,x2,...existed we could change N by defining N'(xt) = yi+1 (i'^ 1), and the resulting matching N' would satisfy y^^e E[D]\N'[D], contradicting the criticality of D). The following slight refinement of the Cantor-Bernstein theorem is well known (see, for example, [2]). THEOREM 1.6. If both B and C are matchable in a bipartite graph (B, C, E) then so is B[) C (that is, the graph has a perfect matching).

2. Transition graphs between bases are matchable In this section we shall prove the following. THEOREM 2.1. If B and C are bases in a matroid M = (S,J) then the transition graph T — (B, C, E) has a perfect matching.

This settles a conjecture made (for the special case of vector spaces) by the second author in [8]. At the end of the section we shall describe the problem which prompted this conjecture, and for which the theorem has an application. If M is a finitedimensional vector space, then the theorem is well known and follows from the fact that the determinant of the transition matrix is non-zero. Proof. Let us first show that C is matchable. Since E(c} is finite for every c e C, it suffices, by M. Hall's theorem [3] to show that \E[A]\ ^ \A\ for every finite subset A of C. But from the definition of the edge set E it is clear that (B\A) U E[A] is spanning. Hence \E[A]\ ^ \A\, by Lemma 1.2. By Theorem 1.6 it suffices now to show that also B is matchable, that is, that F is espousable. Since £ is finite for every ceC, if this is not the case then, by Theorems 1.3 and 1.4, F contains a 1-obstruction (a,D). Let N be a matching of D. Write

F={(b,c):cesc(b)}. We define subsets Xt (i < co) of B as follows: XQ = {a}, Xi+1 = NIFiXJWX*, where X' = (J {Xf.j ^ i). Also define, for i < co, Y{ = N[Xt] and Y* = N[X*\. Note that since F(b} is finite for every b e B, all sets Xt, Yx are finite. We shall call an Af-alternating path xo,y1,x1,y2,x2,...in F admissible if (a) xQ = a, and (b) for every i < co there exists j < co such that x^Xf, and yi+leYm. Let A = {beB:b belongs to some admissible TV-alternating path}. ASSERTION

1. A is finite.

Proof. Define a set K of directed edges in A as follows: if u, v € A then (u, v)eK if uzXp and N(v)e Yj+1 n £. Since each Yi is finite, the out-degree of every vertex in the graph G = (A, K) is finite. By the definition of A, every vertex of A can be reached in G by a directed path from a. Hence, by Konig's lemma, if A is infinite then G contains an infinite path. But this means that there exists an infinite TV-alternating path starting at a. This, by Lemma 1.5, is impossible. 13-2

388

RON AHARONI AND MAURICE POUZET

Let Z = (B\A) U N[A]. ASSERTION

2. A n Xj £ sp (Z) /or eyery _/ <