On the Packing/Covering Conjecture of Infinite Matroids

The Packing/Covering Conjecture was introduced by Bowler and Carmesin motivated by the Matroid Partition Theorem by Edmonds and Fulkerson. A packing for a family $ (M_i: i\in\Theta) $ of matroids on the common edge set $ E $ is a system $ (S_i: i\in\Theta ) $ of pairwise disjoint subsets of $ E $ where $ S_i $ is panning in $ M_i $. Similarly, a covering is a system $ (I_i: i\in\Theta ) $ with $\bigcup_{i\in\Theta} I_i=E $ where $ I_i $ is independent in $ M_i $. The conjecture states that for every matroid family on $ E $ there is a partition $E=E_p \sqcup E_c$ such that $ (M_i \upharpoonright E_p: i\in \Theta) $ admits a packing and $ (M_i. E_c: i\in \Theta) $ admits a covering. We prove the special case where $ E $ is countable and each $ M_i $ is either finitary or cofinitary. The connection between packing/covering and matroid intersection problems discovered by Bowler and Carmesin can be established for every well-behaved matroid class. This makes possible to approach the problem from the direction of matroid intersection. We show that the generalized version of Nash-Williams' Matroid Intersection Conjecture holds for countable matroids having only finitary and cofinitary components.


Introduction
Rado asked in 1966 (see Problem P531 in [24]) if it is possible to extend the concept of matroids to infinite without loosing duality and minors.Based on the works of Higgs (see [19]) and Oxley (see [22] and [23]) Bruhn, Diestel, Kriesell, Pendavingh and Wollan settled Rado's problem affirmatively in [10] by finding a set of cryptomorphic axioms for infinite matroids, generalising the usual independent set-, bases-, circuit-, closure-and rank-axioms for finite mastoids.Higgs named originally these structures B-matroids to distinguish from the original concept.Later this terminology vanished and in the context of infinite combinatorics B-matroids are referred as matroids and the term 'finite matroid' is used to differentiate.
An M = (E, I) is a matroid if I ⊆ P(E) with (1) ∅ ∈ I; (2) I is downward closed; (3) For every I, J ∈ I where J is ⊆-maximal in I but I is not, there exists an e ∈ J \ I such that I + e ∈ I; (4) For every X ⊆ E, any I ∈ I ∩ P(X) can be extended to a ⊆-maximal element of I ∩ P(X).
If E is finite, then (4) is redundant and (1)-( 3) is one of the usual axiomatizations of finite matroids.One can show that every dependent set in an infinite matroid contains a minimal dependent set which is called a circuit.Before Rado's program was settled, a more restrictive axiom was used as a replacement of (4): (4') If all the finite subsets of an X ⊆ E are in I, then X ∈ I.
The implication (4')=⇒(4) follows directly from Zorn's lemma thus axioms (1), ( 2), (3) and (4') describe a subclass F of the the matroids.This F consists of the matroids having only finite circuits and called the class of finitary matroids.Class F is closed under several important operations like direct sums and taking minors but not under taking duals which was the main motivation of Rado's program for looking for a more general matroid concept.The class F * of the duals of the matroids in F consists of the cofinitary matroids, i.e. matroid whose all cocircuits are finite.In order to being closed under all matroid operations we need, we work with the class F ⊕ F * of matroids having only finitary and cofinitary components, equivalently that are the direct sum of a finitary and cofinitary matroid.
Matroid union is a fundamental concept in the theory of finite matroids.For a finite family (M i : i ≤ n) of matroids on a common finite edge set E one can define a matroid i≤n M i on E by letting I ⊆ E be independent in i≤n M i if I = i≤n I i for suitable I i ∈ I M i (see [11]).This phenomenon fails for infinite families of finitary matroids.Indeed, let E be uncountable and let M i be the 1-uniform matroid on E for i ∈ N. Then exactly the countable subsets of E would be independent in i∈N M i and hence there would be no maximal independent set contradicting (4). 1 Even so, Bowler and Carmesin observed (see section 3 in [8]) that the rank formula in the Matroid Partition Theorem by Edmonds and Fulkerson (Theorem 13.3.1 in [15]), namely: can be interpreted in infinite setting via the complementary slackness conditions.In the minimax formula above there is equality for the family (I i : i ≤ n) and partition E = E p E c iff Bowler and Carmesin conjectured that for every family M := (M i : i ∈ Θ) of matroids on a common edge set E there is a family (I i : i ∈ Θ) and partition E = E p E c satisfying the conditions above.To explain the name "Packing/Covering Conjecture" let us provide an alternative formulation.A packing for M is a system (S i : i ∈ Θ) of pairwise disjoint subsets of E where S i is spanning in M i .Similarly, a covering for M is a system (I i : i ∈ Θ) with i∈Θ I i = E where I i is independent in M i .
Conjecture 1.1 (Packing/Covering, Conjecture 1.3 in [8] ).For every family (M i : i ∈ Θ) of matroids on a common edge set E there is a partition E = E p E c in such a way that (M i E p : i ∈ Θ) admits a packing and (M i .E c : i ∈ Θ) admits a covering.
We shall prove the following special case of the Pacing/Covering Conjecture 1.1: Theorem 1.2.For every family (M i : i ∈ Θ) of matroids on a common countable edge set E where It is worth to mention that packings and coverings have a crucial role in other problems as well.For example if (M i : i ∈ Θ) is as in Theorem 1.2 and admits both a packing and a covering, then there is a partition E = i∈Θ B i where B i is a base of M i (see [13]).Maybe surprisingly, the failure of the analogous statement for arbitrary matroids is consistent with set theory ZFC (Theorem 1.5 of [13]) which might raise some scepticism about the provability of Conjecture 1.1 for general matroids.
The Packing/Covering Conjecture 1.1 is closely related to the Matroid Intersection Conjecture which has been one of the central open problems in the theory of infinite matroids: Conjecture 1.3 (Matroid Intersection Conjecture by Nash-Williams, [5]).If M and N are finitary matroids on the same edge set E, then they admit a common independent set I for which there is a partition Aharoni proved in [1] based on his earlier works with Nash-Williams and Shelah (see [3] and [4]) that the special case of Conjecture 1.3 where M and N are partition matroids holds.The conjecture is also known to be true if we assume that E is countable but M and N can be otherwise arbitrary (see [21]).Let us call Generalized Matroid Intersection Conjecture what we obtain from 1.3 by extending it to arbitrary matroids (i.e.omitting the word "finitary").Several partial results has been obtained for this generalization but only for well-behaved matroid classes.The positive answer is known for example if: M is finitary and N is cofinatory [6] or both matroids are singular2 and countable [16] or M is arbitrary and N is the direct sum of finitely many uniform matroids [20].
Bowler and Carmesin showed that their Pacing/Covering Conjecture 1.1 and the Generalized Matroid Intersection Conjecture are equivalent and they also found an important reduction for both (see Corollary 3.9 in [8]).By analysing their proof it is clear that the equivalence can be established if we restrict both conjectures to a class of matroids closed under certain operations.It allows us to prove Theorem 1.4 by showing the following instance of the Generalized Matroid Intersection Conjecture which itself is a common extension of the singular case by Ghaderi [16] and our previous work [21]: Theorem 1.4.If M and N are matroids in F ⊕ F * on the same countable edge set E, then they admit a common independent set I for which there is a partition The paper is organized as follows.In the following section we introduce some notation and fundamental facts about matroids that are mostly well-know for finite ones.In Section 3 we collect some previous results and relatively easy technical lemmas in order be able the discuss later the proof of the main results without any distraction.Then in Section 4 we reduce the main results to a key-lemma.After these preparations the actual proof begins with Section 5 by developing and analysing an 'augmenting path' type of technique.Our main principle from this point is to handle the finitary and the cofinitary parts of matroid N differently in order to exploit the advantage of the finiteness of the circuits and cocircuits respectively.Equipped with these "mixed" augmenting paths we discuss the proof of our key-lemma in Section 6.Finally, we introduce an application in Section 7 about orientations of a graph with in-degree requirements.

Notation and basic facts
In this section we introduce the notation and recall some basic facts about matroids that we will use later without further explanation.For more details we refer to [7].
An enumeration of a countable set X is an N → X surjection that we write as {x n : n ∈ N}.We denote the symmetric difference (X \ Y ) ∪ (Y \ X) of X and Y by X Y .A pair M = (E, I) is a matroid if I ⊆ P(E) satisfies the axioms (1)-( 4).The sets in I are called independent while the sets in P(E) \ I are dependent.An e ∈ E is a loop if {e} is dependent.If E is finite, then (1)-( 3) is one of the the usual axiomization of matroids in terms of independent sets (while (4) is redundant).The maximal independent sets are called bases.If M admits a finite base, then all the bases have the same size which is the rank r(M ) of M otherwise we let r(M ) := ∞. 3 The minimal dependent sets are called circuits.Every dependent set contains a circuit.The components of a matroid are the components of the hypergraph of its circuits.The dual of a matroid M is the matroid M * with E(M * ) = E(M ) whose bases are the complements of the bases of M .For an X ⊆ E, M X := (X, I ∩ P(X)) is a matroid and it is called the restriction of M to X.We write M − X for M (E \ X) and call it the minor obtained by the deletion of X.The contraction of X in M and the contraction of M onto X are M/X := (M * − X) * and M.X := M/(E \ X) respectively.Contraction and deletion commute, i.e., for disjoint X, Y ⊆ E, we have (M/X) − Y = (M − Y )/X.Matroids of this form are the minors of M .The independence of an I ⊆ X in M.X is equivalent with I ⊆ span M * (X \ I).If I is independent in M but I + e is dependent for some e ∈ E \ I then there is a unique circuit C M (e, I) of M through e contained in I + e.We say X ⊆ E spans e ∈ E in matroid M if either e ∈ X or there exists a circuit C e with C − e ⊆ X.By letting span M (X) be the set of edges spanned by X in M , we obtain a closure operation span M : We need some further more subject-specific definitions.From now on let M and N be matroids on a common edge set E. We call a W ⊆ E an (M, N )-wave if M W admits an N.W -independent base.Waves in the matroidal context were introduced by Aharoni and Ziv in [5] but it was also an important tool in the proof of the infinite version of Menger's theorem [2] by Aharoni and Berger.We write cond(M, N ) for the condition: 'For every (M, N )-wave W there is an M -independent base of N.W .'A set I ∈ I M ∩ I N is feasible if cond(M/I, N/I) holds.It is known (see Proposition 3.5) that there exists a ⊆-largest (M, N )-wave which we denote by W (M, N ).Let cond + (M, N ) be the statement that W (M, N ) consists of M -loops and r(N.W (M, N )) = 0.As the notation indicates it is a strengthening of cond(M, N ).Indeed, under the assumption cond + (M, N ), ∅ is an M -independent base of N.W for every wave W .A feasible I is called nice if cond + (M/I, N/I) holds.For X ⊆ E let B(M, N, X) be the (possibly empty) set of common bases of M X and N.X.

Preliminary lemmas and preparation
We collect those necessary lemmas in this section that are either known from previous papers or follow more or less directly from definitions.

Classical results.
The following two statements were proved by Edmonds' in [12]: is independent and spans the same set as I.
Proof.We use induction on m.The case m = 0 is trivial.Suppose that m > 0. On the one hand, the set I − f m + e m is independent and spans the same set as I. On the other hand, C(e j , I − f m + e m ) = C(e j , I) for j < m because f m / ∈ C(e j , I) for j < m.Hence by using the induction hypothesis for I − f m + e m and e 1 , . . ., e m−1 , f 1 , . . ., f m−1 we are done.

Lemma 3.2 (Edmonds' augmenting path method). For I ∈ I M ∩ I N , exactly one of the following statements holds:
(1) There is a partition We will develop in Section 5 a "mixed" augmenting path method which operates differently on the finitary and on the cofinitary part of an N ∈ (F ⊕ F * )(E).The phrase 'augmenting path' refers always to our mixed method except in the proof of Lemma 3.14.Note that E M is an (M, N )-wave witnessed by I M and E N is an (N, M )-wave witnessed by I N .
One can define matroids in the language of circuits (see [10]).The following claim is one of the axioms in that case.For finite matroids the axiom above is demanded only in the special case where X is a singleton (known as "Strong circuit elimination") from which the case of arbitrary X can be derived by repeated application.Proof.For every x ∈ C \ I we pick a circuit C x with C x \ I = {x}.If e ∈ C x for some x, then f := x is as desired.Suppose for a contradiction that there is no such an x.Then by Circuit elimination (Claim 3.3) we obtain a circuit through e which is contained entirely in I contradicting the independence of I.
The following statement was shown by Aharoni and Ziv in [5] using a slightly different terminology.

Proposition 3.5. The union of arbitrary many waves is a wave.
Proof.Suppose that W β is a wave for β < κ and let W <α := β<α W β for α ≤ κ.We fix a base B β ⊆ W β of M W β which is independent in N.W β .Let us define B <α by transfinite recursion for α ≤ κ as follows.
First we show by transfinite induction that B <α is spanning in M W <α .For α = 0 it is trivial.For a limit α it follows directly from the induction hypothesis.
The independence of B <α in N.W <α can be reformulated as "W <α \ B <α is spanning in N * W <α ", which can be proved the same way as above.
3.2.Some more recent results and basic facts.Theorem 3.6 (Aigner-Horev, Carmesin and Frölich; Theorem 1.5 in [6]).If M ∈ F(E) and N ∈ F * (E), then there is an Proof.Let E M , E N , I M and I N be as in Theorem 3. 6.Then E M is a wave witnessed by I M thus by cond + (M, N ) we know that E M consists of M -loops and r(N.E M ) = 0.But then I M = ∅ and I N is a base of N which is independent in M .
Observation 3.12.cond + (M, N ) implies cond + (M X, N.X) for every X ⊆ E. Proposition 3.13.If I 0 ∈ I N ∩ I M and I 1 is feasible with respect to (M/I 0 , N/I 0 ), then I 0 ∪ I 1 is feasible with respect to (M, N ).If in addition I 1 is a nice feasible in regards to (M/I 0 , N/I 0 ), then so is Proof.Let W be an (M/e, N/e)-wave.Note that (N/e).W = N.W by definition.Pick a B ⊆ W which is an N.W -independent base of (M/e) W .We may assume that e ∈ span M (W ) and e is not an M -loop.Indeed, otherwise (M/e) W = M W holds and hence W is also an (M, N )-wave.Thus by cond + (M, N ) we may conclude that W consists of M -loops and hence B = ∅, moreover, r(N.W ) = 0 and therefore ∅ is a base of N.W .
Then B is not a base of M W but "almost", namely r(M/B W ) = 1.We apply the augmenting path Lemma 3.2 by Edmonds with B in regards to M W and N.W .An augmenting path P cannot exist.Indeed, if P were an augmenting path then B P would show that W is an (M, N )-wave which does not consist of M -loops, contradiction.Thus we get a partition W = W 0 W 1 instead where W 0 is an (M W, N.W )-wave witnessed by B ∩ W 0 , and therefore also an (M, N )-wave, and W 1 is an (N.W, M W )-wave showed by B ∩ W 1 .Then W 0 must consist of M -loops by cond + (M, N ) and therefore B ⊆ W 1 by the M -independence of B. We need to show that B is spanning not just in N.W 1 but also in N.W .To do so let B be a base of spanning in N because r(N.W 0 ) = 0 by Observation 3.8.We conclude that B is spanning in N.W as desired.
Thus the sets that are witnessing that W \ L is an (M − L, N − L, W \ L)-wave are exactly those that are witnessing that W is an (M, N )-wave, moreover, Proof.The matroids M/X 0 − Y 0 and M/X 1 − Y 1 are the same as well as the matroids N/X 0 − Y 0 and N/X 1 − Y 1 .

Reductions
We repeat here our main results for convenience: Theorem 1.2.For every family (M i : i ∈ Θ) of matroids on a common countable edge set E where M i ∈ F ⊕ F * , there is a partition E = E p E c such that (M i E p : i ∈ Θ) admits a packing and (M i .E c : i ∈ Θ) admits a covering.Theorem 1.4.If M and N are matroids in F ⊕ F * on the same countable edge set E, then they admit a common independent set I for which there is a partition 4.1.Matroid Intersection with a finitary M .As we mentioned, the method by Bowler and Carmesin used to prove Corollary 3.9 in [8] works not only for the class of all matroids but can be adapted for every class closed under certain operations.We apply their technique to obtain the following reduction: Proof.First we show that one can assume without loss of generality in the proof of Theorem 1.2 that Θ is countable.To do so let We apply Theorem 1.2 with E and with the countable family (M i E : i ∈ Θ ).Then we obtain a partition E = E p E c such that (M i E p : i ∈ Θ ) admits a packing (S i : i ∈ Θ ) and (M i E .E c : i ∈ Θ ) admits a covering (I i : i ∈ Θ ).Let E p := E p and and I i := ∅ otherwise.Then (I i : i ∈ Θ) is a covering for (M i .E c : i ∈ Θ) and we are done.We proceed with the proof of Theorem 1.2 assuming that Θ is countable.For i ∈ Θ, let M i be the matroid on E × {i} that we obtain by "copying" M i via the bijection e → (e, i).Then for Then for each e ∈ E c there must be some i ∈ Θ with (e, i) ∈ I N because E N ⊆ span N (I N ).Thus the sets J i cover E c and so do the sets Then {f } × Θ ⊆ E M follows directly from the definition of E p , in particular (f, i) ∈ E M .We know that (f, i) ∈ span M (I M ) and hence f ∈ span M i ({e ∈ E : (e, i) ∈ I M }).Suppose for a contradiction that h ∈ E c ∩ {e ∈ E : (e, i) ∈ I M }.Then for some j ∈ Θ we have (h, j) ∈ E N .Since (h, j) ∈ span N (I N ), we have (h, k) ∈ I N for some k ∈ Θ.But then (h, i), (h, k) ∈ I are distinct elements thus i = k which contradict the N -independence of I. Therefore E c ∩ {e ∈ E : (e, i) ∈ I M } = ∅.Since {e ∈ E : (e, i) ∈ I M } ⊆ J i by the definition of J i we conclude {e ∈ E : (e, i) ∈ I M } = S i .Therefore (S i : i ∈ Θ) is a packing for (M i E p : i ∈ Θ) and (I i : i ∈ Θ) is a covering for (M i .E c : i ∈ Θ) as desired.Now we derive Theorem 1.4 from Theorem 1.2.To do so, we take a partition E p and we define I N := R M .By construction E p ⊆ span M (I M ) and I N ∈ I M.Ec .We also know that

4.2.
Finding an M -independent base of N .The following reformulation of the matroid intersection problem was introduced by Aharoni and Ziv in [5] but its analogue by Aharoni was already an important tool to attack (and eventually solve in [2]) the Erdős-Menger Conjecture.Corollary 3.11).By Claim 4.2, there is an M/E M -independent base

Reduction to a key-lemma.
From now on we assume that M ∈ F(E) and N ∈ (F ⊕ F * )(E) where E is countable.Let E 0 be the union of the finitary components of N and let E 1 := E \ E 0 .Note that N E 0 is finitary, N E 1 is cofinitary and no N -circuit meets both E 0 and E 1 .way that e n ∈ span N (I n+1 ).Suppose that I n is already defined.If e n / ∈ span N (I n ), then we apply Lemma 4.4 with (M/I n , N/I n ) and e n and take the union of the resulting I with I n to obtain I n+1 (see Observation 3.13), otherwise let I n+1 := I n .The recursion is done.Now we construct an M -independent Thus by Corollary 3.7 there is an M/I n -independent base B n of (N E 1 )/(E 1 ∩ I n ) and Let U be a free ultrafilter on P(N) ad we define Then S is M -independent and N -spanning and therefore we are done.Indeed, suppose for a contradiction that S contains an M -circuit C. For e ∈ C, we pick a U e ∈ U with e ∈ I + n for n ∈ U e .Since M is finitary, C is finite, thus U := ∩{U e : e ∈ C} ∈ U.But then for n ∈ U we have C ⊆ I + n which contradicts the M -independence of I + n .Clearly, I n+1 ⊆ S for every n ∈ N and therefore E 0 ⊆ span N (S).Finally, suppose for a contradiction that there is some

Mixed augmenting paths
In the section we introduce an 'augmenting path' type of method and analyse it in order to show some properties we need later.On E 0 the definition will be the same as in the proof of the Matroid Intersection Theorem by Edmonds [12] but on E 1 we need to define it in a different way considering that N E 1 is cofinitary.For brevity we write Lemma 5.4.If I is a nice dually safe feasible set and P = {x 1 , . . ., x 2n+1 } is an augmenting path for I, then I P can be extended to a nice dually safe feasible set.
The sets X 0 := I + x 2n+1 and X 1 := I P have the same M -span (see Proposition 5.3/ (A)).Recall that M = M/X 1 and N = N/X 1 by definition.Hence Lemma 3.20 ensures We have cond + (M/I, N/I) because I is a nice feasible set by assumption.Then by Observation 3.17 This means by definition that x has the same out-neighbours in D(I) and D(I P ).
We turn to the case where x ∈ I 0 .By definition C N (y, I) is well-defined and contains x, in particular y ∈ E 0 .For k ≤ n, let us denote I +x 1 −x 2 +x 3 −. ..−x 2k +x 2k+1 by I k .Note that I n = I P .We show by induction on k that I k is N -independent and x ∈ C N (y, I k ).Since I + x 1 is N -independent by definition and x 1 = y because y / ∈ P by assumption, we obtain C N (y, I) = C N (y, I 0 ), thus for k = 0 it holds.Suppose that n > 0 and we already know the statement for some k < n.We have x 2k+2 because there is no "jumping arc" in the augmenting path by property (iv).It follows via the N -independence of I k that I k+1 is also N -independent.
Finally, for y ∈ E(x, n), P n does not contain y or any of its out-neighbours with respect to D(I n ) because P n ∩ E(x, n) = ∅.Hence by applying Lemma 5.5 with P n , y and I n (and then Observation 5.6) we may conclude that yz ∈ D(I n+1 ) whenever yz ∈ D(I n ).This implies E(x, n) ⊆ E(x, n + 1) since reachability from x is witnessed by the same directed paths. Let We are going to show that B witnesses that W is a wave.Since M is finitary the M -independence of the sets 1) and ( 2) of Claim 6.1 ensure W ⊆ span M (B), while (1) and (3) guarantee B 1 ⊆ span N * (W \ B).The latter means that B 1 is independent in N.(W 1 \ B 1 ).Suppose for a contradiction that B 0 is not independent in N.W 0 .Then there exists an N -circuit C ⊆ E 0 that meets B but avoids W \ B. We already know that B 0 is N -independent thus C is not contained in and hence B in N.W as well therefore W is a wave.
By cond + (M, N ) we know that W consists of M -loops and r(N.W ) = 0.It implies r(N.X) = 0 because X ⊆ W by definition.This means X ⊆ span N (E \ X).Since X ⊆ E 0 and E 0 is the union of the finitary N -components, X ⊆ span N (E 0 \ X) follows.Thus for every x ∈ X there is a finite N -circuit C ⊆ E 0 with C ∩ X = {x}.The sequence (span N (I n ) ∩ E 0 ) is ascending by construction and exhausts E 0 \ X by the definition of X.As C − x ⊆ E 0 \ X is finite, this implies that for every large enough n, I n spans C − x in N and hence spans x itself as well.But then by the definition of X, we must have X = ∅.Therefore (span N (I n ) ∩ E 0 ) exhausts E 0 and the proof of Lemma 4.4 is complete.
7. An application: Degree-constrained orientations of infinite graphs Matroid intersection is a powerful tool in graph theory and in combinatorial optimization.Our generalization Theorem 1.4 extends the scope of its applicability to infinite graphs.
To illustrate this, let us consider a classical problem in combinatorial optimization.A graph is given with degree-constrains and we are looking for either an orientation that satisfies it or a certain substructure witnessing the non-existence of such an orientation (see [17]).
Let a (possibly infinite) graph G = (V, E) be fixed through this section.We denote the set of edges incident with v by δ(v).Let o : V → Z with |o(n)| ≤ d(v) for v ∈ V which we will threat as 'lower bounds' for in-degrees in orientations in the following sense.We say that the orientation D of G is above o at v if either o(v) ≥ 0 and v has at least o(v) ingoing edges in D or o(v) < 0 and all but at most −o(v) edges in δ(v) are oriented towards v by D. We say strictly above if we forbid equality in the definition.Orientation D is above o if it is above o at every v ∈ V .We say that D is (strictly for which I M contains a base of M v and let V := V \ V .We claim that whenever an edge e ∈ E is incident with some v ∈ V , then I contains one of a e and a e .Indeed, if I contains none of them then they cannot be N -spanned by I N thus they are M -spanned by M which implies that both end-vertices of e belong to V , contradiction.Thus if e is incident some v ∈ V , then all ingoing arcs of v in D must be in I. Then the M -independence of I ensures that D is bellow o at v. Suppose for a contradiction that a e is an arc in D from a v ∈ V to a w ∈ V .As we have already shown, we must have a e ∈ I .By w ∈ V we know that I M contains a base of M w thus a e / ∈ I N by the M -independence of I and therefore a e ∈ I M .But then a e cannot be spanned by I N in N hence a e ∈ span M (I M ), which means that I M contains a base of M v contradicting v ∈ V .We conclude that all the edges between V and V are oriented towards V in D. By the definition of V , D is above o at every w ∈ V .If D is also above o for every v ∈ V , then D is above o.Otherwise there exists a v ∈ V such that D is strictly bellow o at v, but then V is as desired.
Easy calculation shows that if G is finite, then the existence of a V described in Theorem 7.1 implies the non-existence of an orientation above o.Indeed, the total demand by o on V is more than the number of all the edges that are incident with a vertex in V .That is why for finite G, "if" can be replaced by "if and only if" in Theorem 7.1.For an infinite G it is not always the case.Indeed, let G be the one-way infinite path v 0 , v 1 , . . .and let o(v n ) = 1 for n ∈ N. Then orienting edge {v n , v n+1 } towards v n+1 for each n ∈ N and taking V := V satisfies the three points in Theorem 7.1.However, taking the opposite orientation is above o.
A natural next step is to introduce upper bounds p : V → Z beside the lower bounds o : V → Z.To avoid trivial obstructions we assume that o and p are consistent which means that for every v ∈ V there is an orientation D v which is above o at bellow p at v. Question 7.2.Let G be a countable graph and let o, p : V → Z be a consistent pair of bounding functions.Suppose that there are orientations D o and D p that are above o and bellow p respectively.Is there always a single orientation D which is above o and bellow p?
The positive answer for finite graphs is not too hard to prove, as far we know its first appearance in the literature is [14].

Claim 3 . 3 (
Circuit elimination axiom).Assume that C e is a circuit and {C x : x ∈ X} is a family of circuits where X ⊆ C − e and C x is a circuit with C ∩ X = {x} avoiding e.Then there is a circuit through e contained in C ∪ x∈X C x \ X =: Y Proof.Since C x −x spans x we have C−e ⊆ span(Y −e) and therefore e ∈ span(span(Y −e)).But then e ∈ span(Y − e) because span is a closure operator.

Corollary 3 . 4 .
Let I be an independent and suppose that there is a circuit C ⊆ span(I) with e ∈ I ∩ C. Then there is an f ∈ C \ I with e ∈ C(f, I).

Proof.Lemma 3 . 14 .
By definition the feasibility of I 1 w.r.t.(M/I 0 , N/I 0 ) means that the condition cond(M/(I 0 ∪ I 1 ), N/(I 0 ∪ I 1 )) holds.The feasibility of I 0 ∪ I 1 w.r.t.(M, N ) means the same also by definition.For 'nice feasible' the argument is similar, only cond must be replaced by cond + .The following lemma was introduced in[21].Condition cond + (M, N ) implies that whenever W is an (M/e, N/e)-wave for some e ∈ E witnessed by B ⊆ W , then B ∈ B(M/e, N/e, W ), i.e.B is spanning in N.W .

3. 3 .Proposition 3 . 15 .
Technical lemmas in regards to B(M, N, W ). For W := W (M, N ), the elements of B(M, N, W ) are nice feasible sets.Proof.Let B ∈ B(M, N, W ). Clearly B ∈ I M ∩ I N because B ∈ I M ∩ I N.W by definition and I N.W ⊆ I N .We know that W \ B is an (M/B, N/B)-wave consisting of M/B loops and r(N.(W \ B)) = 0 because B is a base of N.W .In order to show cond + (M/B, N/B), it is enough to prove that W \ B is actually W (M/B, N/B) =: W . Suppose for a contradiction that W W \ B. Let B be an

Observation 3 . 17 .Observation 3 . 18 .Lemma 3 . 19 .
wave witnessed by B ∪ B which contradicts the maximality of W . Corollary 3.16.Assume that I ∈ I M ∩ I M and B ∈ B(M/I, N/I, W ) where W := W (M/I, N/I).Then I ∪ B is a nice feasible set.Proof.Combine Propositions 3.13 and 3.15.If cond + (M, N ) holds and L is a set of M -loops, then cond + (M − L, N − L) also holds.Let W be a wave and let L be a set of common loops of M and N .Then W \ L is also a wave and B(M, N, W ) = B(M, N, W \ L).Let W be a wave and L ⊆ W such that L consists of M -loops with r(N.L) = 0. Then W \ L is an (M − L, N − L, W \ L)-wave with

Lemma 4 . 1 .
Theorems 1.2 and 1.4 are implied by the special case of Theorem 1.4 where M ∈ F.

Claim 4 . 2 .Lemma 4 . 3 .
Assume that M ∈ F(E) and N ∈ (F ⊕ F * )(E) such that E is countable and cond + (M, N ) holds.Then there is an M -independent base of N .Claim 4.2 implies our main results Theorems 1.2 and 1.4.Proof.By Lemma 4.1 it is enough to show that the special case of Theorem 1.4 where M ∈ F follows from Claim 4.2.To do so, let E M := W (M, N ) and let

Lemma 4 . 4 Lemma 4 . 5 .
(key-lemma).If cond + (M, N ) holds, then for every e ∈ E 0 there is a nice feasible I with e ∈ span N (I).Lemma 4.4 implies our main results Theorems 1.2 and 1.4.Proof.It is enough to show that Lemma 4.4 implies Claim 4.2 because of Lemma 4.3.Let us fix an enumeration {e n : n ∈ N} of E 0 .We build an ⊆-increasing sequence (I n ) of nice feasible sets starting with I 0 := ∅ (who is nice feasible by cond + (M, N )) in such a Proof.Let M := M/(I P ) and N := N/(I P ).By Lemma 5.1 it is enough to show that B(M , N , W ) = ∅ for W := W (M , N ).Statements (A) and (B) of Proposition 5.3 ensure that the elements of L := {x 1 , x 3 , . . ., x 2n−1 } ∩ E 0 are common loops of M and N .Statement (C) tells that Y 0 := • span M (I) 1 and Y 1 := • span M (I) 1 P 1 have the same N * -span, furthermore, Y 1 is N * -independent, i.e. r(N.Y 1 ) = 0. Note that Y 1 consists of M -loops by (A).Thus by applying Observation 3.18 with W and L and then Lemma 3.19 with W \ L and Y 1 we can conclude that W \

Lemma 5 . 5 .
cond + (M/I − Y 0 , N/I − Y 0 ) also holds.Applying Lemma 3.14 withM/I−Y 0 , N/I−Y 0 , W \(Y 0 ∪L) and x 2n+1 tells B(M/X 0 −Y 0 , M/X 0 −Y 0 , W \(Y 1 ∪L)) = ∅ which completes the proof.If P = {x 1 , . . ., x 2n+1 } isan augmenting path for I which contains neither x nor any of its out-neighbours in D(I), then xy ∈ D(I) implies xy ∈ D(I P ).Proof.Suppose that xy ∈ D(I).First we assume that x / ∈ I. Then the set of the out-neighbours of x is C M (x, I) − x.By assumption P ∩ C M (x, I) = ∅ and therefore C M (x, I) ⊆ I P thus C M (x, I) = C M (x, I P ).This means by definition that x has the same out-neighbours in D(I) and D(I P ).If x ∈ I 1 , then we can argue similarly.The set of the out-neighbours of x in D(I) is C N * (x, • span M (I) 1 ) − x.By assumption P ∩ C N * (x, • span M (I) 1 ) = ∅ and therefore C N * (x, • span M (I) 1 ) ∩ (I P ) = {x}.Since span 1 M (I P ) ⊇ span 1 M (I) because of Proposition 5.3/(A), we also have C N * (x, • span M (I) 1 ) ⊆ span 1 M (I P ).By combining these we conclude B. Hence C \ B = C \ W = ∅.Let us pick some e ∈ C ∩ B. Since C is finite, for every large enough n we have C ∩ B ⊆ C ∩ I n and I n spans C in N (for the latter we use X ⊆ W \ B).Applying Corollary 3.4 with I n , N, C and e tells that e ∈ C N (f, I n ) for some f ∈ C \ W whenever n is large enough.Then by (5) of Claim 6.1 we can take an x ∈ X and an n Finally, D is (strictly) bellow o if the reverse of D is strictly above −o.Theorem 7.1.Let G = (V, E) be a countable graph and let o : V → Z.If there is no orientation of G above o, then there is a V ⊆ V and an orientation D of G such that• D is bellow o at every v ∈ V ; • There exists a v ∈ V such that D is strictly bellow o at v; • Every edge between V and V \ V is oriented by D towards V .Proof.Without loss of generality we may assume that G is loopless.We define the digraph ↔ G = (V, A) by replacing each e ∈ E by back and forth arcs a e , a e between the end-vertices of e.Let δ + (v) be the set of the ingoing edges of v in↔ G.For v ∈ V , let M v be U δ + (v),o(v) if o(v) ≥ 0 and U * δ + (v),−o(v) if o(v) < 0.We define N e to be U {ae,a e },1 for e ∈ E. Let M := v∈V M v and N := e∈E N e .Since M, N ∈ (F ⊕ F * )(A), Theorem 1.4 guarantees that there exists an I ∈ I M ∩ I N and a partition A = A M A N such that I M := I ∩ A M spans A M in M and I N := I ∩ A N spans A N in N .Note that |I ∩ {a e , a e }| ≤ 1 by the N -independence of I. We define D by taking the orientation a e of e if a e ∈ I and a e otherwise.Let V consists of those vertices v If x 2k+2 / ∈ C N (y, I k ) then Proof.If y ∈ E(x, n) \ I n , then its out-neighbours in D(I n ) are in E(x, n) ∩ I n and span y in M .Thus I n+1 \ (I n P n ) cannot contain any edge from E(x, n).I n ∩ E(x, n) = I n+1 ∩ E(x, n) and for every y ∈ E(x, n) \ I n we have C M (y, I n ) = C M (y, I n+1 ) ⊆ E(x, n).For y ∈ E(x, n) \ I n ,y has the same out-neighbours in D(I n ) and in D(I n+1 ) and they span y in N * .More concretely: