Countable Menger theorem with finitary matroid constraints on the ingoing edges

We present a strengthening of the countable Menger theorem (edge version) of R. Aharoni. Let $ D=(V,A) $ be a countable digraph with $ s\neq t\in V $ and let $\mathcal{M}=\bigoplus_{v\in V}\mathcal{M}_v $ be a matroid on $ A $ where $ \mathcal{M}_v $ is a finitary matroid on the ingoing edges of $ v $. We show that there is a system of edge-disjoint $ s \rightarrow t $ paths $ \mathcal{P} $ such that the united edge set of the paths is $ \mathcal{M} $-independent, and there is a $ C \subseteq A $ consists of one edge from each element of $ \mathcal{P} $ for which $ \mathsf{span}_{\mathcal{M}}(C) $ covers all the $ s\rightarrow t $ paths in $ D $.


Introduction
In this paper we generalize the countable version of Menger's theorem of Aharoni [1] by applying the results of Lawler and Martel about polymatroidal flows (see [5]).
Theorem 1 (Menger). Let D = (V, A) be a finite digraph with s = t ∈ V . Then the maximum number of the pairwise edge-disjoint s → t paths is equal to the minimal number of edges that cover all the s → t paths.
Erdős observed during his school years that the theorem above remains true for infinite digraphs (by saying cardinalities instead of numbers). He felt that this is not the "right" infinite generalization of the finite theorem and he conjectured the "right" generalization which was known as the Erdős-Menger conjecture. It is based on the observation that in Theorem 1 an optimal cover consists of one edge from each path of an optimal path-system. The Erdős-Menger conjecture states that for arbitrary large digraphs there is a path-system and a cover that satisfy these complementarity conditions. After a long sequence of partial results the countable case has been settled affirmatively by R. Aharoni: Theorem 2 (R. Aharoni, [1]). Let D = (V, A) be a countable digraph with s = t ∈ V . Then there is a system P of edge-disjoint s → t paths such that there is a edge set C that covers all the s → t paths in D and C consists of choosing one edge from each P ∈ P.
It is worth to mention that R. Aharoni and E. Berger proved the Erdős-Menger conjecture in its full generality in 2009 (see [2]) which was one of the greatest achievements in the theory of infinite graphs. We present the following strengthening of the countable Menger's theorem above. Then there is a system of edge-disjoint s → t paths P such that the united edge set of the paths is M-independent, and there is an edge set C consists of one edge from each element of P for which span M (C) covers all the s → t paths in D.
Instead of dealing with covers directly, we are focusing on t − s cuts (X ⊆ V is a t − s cut if t ∈ X ⊆ V \ {s}). Let us call a path-system P independent if A(P) is independent in M. Suppose that an independent system P of edge-disjoint s → t paths and a t − s cut X satisfy the complementarity conditions: Then clearly P and C := A(P) ∩ in D (X) satisfy the demands of Theorem 3. Therefore it is enough to prove the following reformulation of the theorem.

Main result
Theorem 5. Let D = (V, A) be a countable digraph with s = t ∈ V and suppose that there is a finitary matroid M v on in D (v) for each v ∈ V and let M = v∈V M v . Then there is a system P of edge-disjoint s → t paths where A(P) is independent in M and a t − s cut X such that P and X satisfy the complementarity conditions (Condition 4).
Proof. Without loss of generality we may assume that M does not contain loops. A pair (W, X) is called a wave if X is a t − s cut and W is an independent system of edge-disjoint s → X paths such that the second complementarity condition holds for W and X (i.e. A last (W) spans in D (X) in M).
Remark 6. By picking an arbitrary base B of out(s) and taking W := B as a set of single-edge paths and X := V \ {s} we obtain a wave (W, X) thus always exists some wave.
We say that the wave (W 1 , X 1 ) extends the wave (W 0 , X 0 ) and write 2. W 1 consists of the forward continuations of some of the paths in W 0 such that the continuations lie in X 0 , 3. W 1 contains all of those paths of W 0 that meet X 1 .
If in addition W 1 contains a forward-continuation of all the elements of W 0 , then the extension is called complete. Note that ≤ is a partial ordering on the waves and if (W 0 , X 0 ) ≤ (W 1 , X 1 ) holds, then the extension is proper (i.e. Observation 7. If (W 1 , X 1 ) is an incomplete extension of (W 0 , X 0 ), then it is a proper extension thus X 1 X 0 . Furthermore, W 1 and X 0 do not satisfy the second complementarity condition (Condition 4/2).
Lemma 8. If a nonempty set X of waves is linearly ordered by ≤, then X has a unique smallest upper bound sup(X ).
Proof: We may suppose that X has no maximal element. Let (W ξ , X ξ ) : ξ < κ be a cofinal sequence of (X , ≤). We define X := ξ<κ X ξ and The paths in W are pairwise edge-disjoint since P 1 , P 2 ∈ W implies that P 1 , P 2 ∈ W ξ for all large enough ξ. Since the matroid M is finitary the same argument shows that W is independent.
Suppose that e ∈ in D (X) \ A(W). For a large enough ξ < κ we have e ∈ in D (X ξ ). Then the last edges of those elements of W ξ that terminate in head(e) spans e in M. These paths have to be elements of all the further waves of the sequence (because of the definition of ≤) and thus of W as well. Therefore (W, X) is a wave and clearly an upper bound.
Suppose that (Q, Y ) is another upper bound for X . Then X ξ ⊇ Y for all ξ < κ and hence X ⊇ Y . Let Q ∈ Q be arbitrary. We know that W ξ contains an initial segment Q ξ of Q for all ξ < κ because (Q, Y ) is an upper bound (see the definition of ≤). For ξ < ζ < κ the path Q ζ is a (not necessarily proper) forward continuation of Q ξ . From some index the sequence Q ξ : ξ < κ need to be constant, say Q * , since Q is a finite path. But then Q * ∈ W. Thus any Q ∈ Q is a forward continuation of a path in W. Finally assume that some P ∈ W meets Y . Pick a ξ < κ for which P ∈ W ξ . Then The Remark 6 and Lemma 8 imply via Zorn's Lemma the following.
Corollary 9. There exists a maximal wave. Furthermore, there is a maximal wave which is greater or equal to an arbitrary prescribed wave.
Let (W, X) be a maximal wave. To prove Theorem 5 it is enough to show that there is an independent system of edge-disjoint s → t paths P that consists of the forward-continuation of all the paths in W. Indeed, condition A(P) ∩ out D (X) = ∅ will be true automatically (otherwise P would violate independence, when the violating path "comes back" to X) and hence P and X will satisfy the complementarity conditions.
We need a method developed by Lawler and Martel in [5] for the augmentation of polymatroidal flows in finite networks which works in the infinite case as well.
Lemma 10. Let P be an independent system of edge-disjoint s → t paths. Then there is either an independent system of edge-disjoint s → t paths P ′ with span Mt (A last (P)) span Mt (A last (P ′ )) or there is a t − s cut X such that the complementarity conditions (Condition 4) hold for P and X. If there is an augmenting walk terminating in t, then let W be a shortest such a walk. Build P ′ from the edges A(W )△A(P) in the following way. Keep untouched those P ∈ P for which A(W ) ∩ A(P ) = ∅ and replace the remaining finitely many paths, say Q ⊆ P where |Q| = k, by k + 1 new s → t paths constructed from the edges A(W )△A(Q) by the greedy method. Obviously P ′ is an independent system of edge-disjoint s → t paths. We need to show that span Mt (A last (P)) span Mt (A last (P ′ )).
If only the last vertex of W is t, then it is clear. Let f 1 , e 1 , . . . , f n , e n , f n+1 be the edges of W incident with t enumerated with respect to the direction of W . The initial segments of W up to the inner appearances of t may not be augmenting walks (since W is a shortest that terminates in t) hence by condition 4 the one edge longer and the one edge shorter segments are. It follows that for any 1 ≤ i ≤ n there is a M t -circuit C i in Furthermore, f i / ∈ A(P) and e i ∈ C i ∩ A(P). It implies by induction that A i \ {e i } spans the same set in M t as A(P) ∩ in D (t) whenever 1 ≤ i ≤ n and hence A n ∪ {f n+1 } spans a strictly larger.
Suppose now that none of the augmenting walks terminates in t. Let us denote the set of the last vertices of the augmenting walks by Y . We show that P and X := V \Y satisfy the complementarity conditions. Obviously X is a t − s cut. Suppose, to the contrary, that e ∈ A(P) ∩ out D (X). Pick an augmenting walk W terminating in head(e). Necessarily e ∈ A(W ), otherwise W e would be an augmenting walk contradicting to the definition of X. Consider the initial segment W ′ of W for which the following edge is e. Then W ′ e is an augmenting walk (if W ′ itself is not, then it is because of condition 4) which leads to the same contradiction.
To show the second complementarity condition assume that f ∈ in D (X) \ A(P). Choose an augmenting walk W that terminates in tail(f ). We may suppose that f / ∈ A(W ) otherwise we consider the initial segment W ′ of W for which the following edge is f (it is an augmenting walk, otherwise W ′ f would be by applying condition 4). The initial segments of W f that terminate in head(f ) may not be augmenting walks. Let f 1 , e 1 , . . . , f n , e n be the ingoing-outgoing edge pairs of head(f ) in W with respect to the direction of W (enumerating with respect to the direction of W ) and let f n+1 := f . Then for any 1 ≤ i ≤ n + 1 there is a unique M-circuit C i in It follows by using condition 4 and the definition of X that for 1 ≤ i ≤ n Assume that we already know for some 1 ≤ i ≤ n that f j is spanned by F := A(P) ∩ in D (X) in M whenever j < i. Any element of C i \ {f i } which is not in F has a form f j for some j < i thus by the induction hypothesis it is spanned by F and hence we obtain that f i ∈ span M (F ) as well. By induction it is true for i = n + 1.
Proposition 11. Assume that (W, X) and (Q, Y ) are waves where Y ⊆ X and Q consists of the forward-continuation of some of the paths in W where the new terminal segments lie in X. Let Proof: The path-system W Y ∪ Q (not necessarily disjoint union) is edge-disjoint since the edges in A(Q) \ A(W) lie in X. For the same reason it may violate independence only at the vertices It is routine to check that the choice Observation 12. If (W, X) is a wave and for some A 0 ⊆ A\A(W) the set A 0 ∪A(W) is independent, then (W, X) is a D(A 0 )-wave as well.
Lemma 13. If (W, X) is a maximal D-wave and e ∈ A\A(W) for which A(W)∪{e} is independent, then all the extensions of the D(e)-wave (W, X) in D(e) are complete.
Proof: Seeking a contradiction, assume that we have an incomplete extension (Q, Y ) of (W, X) with respect to D(e). Observe that necessarily e ∈ in D (Y ) and r M (in D (Y )/A last (Q)) = 1. Furthermore, Y X by Observation 7. We show that (W, X) has a proper extension with respect to D as well contradicting to its maximality. Without loss of generality we may assume that in D (X) = A last (W). Indeed, otherwise we delete the edges in D (X) \ A(W) from D and from M. It is routine to check that after the deletion (W, X) is still a wave and a proper extension of it remains a proper extension after putting back these edges.
Contract V \ X to s and contract Y to t in D and keep M unchanged. Apply the augmenting walk method (Lemma 10) in the resulting system with the V \X → Y terminal segments of the paths in Q. If the augmentation is possible, then the assumption in D (X) = A last (W) ensures that the first edge of any element of the resulting path-system R is a last edge of some path in W. By uniting the elements of R with the corresponding paths from W we can get a new independent system of edge-disjoint s → Y paths Q ′ (with respect to D). Furthermore, r M (in D (Y )/A last (Q)) = 1 guarantees that A last (Q ′ ) spans in D (Y ) in M and hence (Q ′ , Y ) is a wave. Thus by Proposition 11 we get an extension of (W, X) and it is proper because Y X which is impossible. Thus the augmentation must be unsuccessful which implies by Lemma 10 that there is some Z with Y ⊆ Z ⊆ X such that Z and Q satisfy the complementarity conditions. By Proposition 7 we know that Z X. For the initial segments Q Z of the paths in Q up to Z the pair (Q Z , Z) forms a wave. Thus by applying Proposition 11 with (W, X) and (Q Z , Z) we obtain an extension of (W, X) which is proper because Z X contradicting to the maximality of (W, X).
Proof: It is equivalent to show that there exists a v → t path Q in D − span M (A(W )) (path Q will necessarily lie in D[X] because D − span M (A(W )) does not contain any edge entering into X.) Suppose, to the contrary, that it is not the case. Let X ′ X be the set of those vertices in X that are unreachable from v in D − span M (A(W )) (note that v / ∈ X ′ but t ∈ X ′ by the indirect assumption). Let W ′ be consist of the paths in W that meet X ′ . If we prove that (W ′ , X ′ ) is a wave, then we are done since it would be a proper extension of the maximal wave (W, X). Assume that f ∈ in D (X ′ ) \ A(W ′ ). Then by the definition of X ′ we have tail(f ) ∈ V \ X thus f ∈ in D (X). Hence f is spanned by the last edges of the paths in W terminating in head(f ) and all these paths are in W ′ as well. Therefore (W ′ , X ′ ) is a wave.
Lemma 15. Let (W, X 0 ) be a maximal wave and assume that P ∈ W and let W 0 = W \ {P }. Then there is an s-arborescence A such that

there is a maximal wave with respect to D(A(A)) which is a complete extension of the D(A(A)
)-wave (W 0 , X 0 ).
Proof: It is clearly a wave thus we show just the maximality. Suppose that (Q, Y ) is a proper extension of (W \ {P }, X 0 ) with respect to D(A(P )). Necessarily end(P ) ∈ Y otherwise it would be a wave with respect to D which properly extends (W, X 0 ). Let e be the last edge of P . We know that But then it properly extends (W, X 0 ) which is a contradiction.
Fix a well-ordering of A with order type |A| ≤ ω. We build the arborescence A by recursion. Let A 0 := P . Assume that A m , W m and X m has already defined for m ≤ n in such a way that It may not be a wave with respect to D. Hence the s-arborescence A ∞ contains an edge e ∈ in D (X ∞ ). Apply Proposition 14 with (W ∞ , X ∞ ) and head(e) in the system D(A(A ∞ )). Consider the last vertex v of the resulting Q which is in V (A ∞ ). Since v = t by assumption there is an outgoing edge f of v in Q. Then f ∈ out D−span M (A(W∞)) (V (A ∞ )) which implies that for some n 0 < ω we have f ∈ out D−span M (A(Wn)) (V (A n )) whenever n > n 0 . But then the infinitely many pairwise distinct edges {e n : n 0 < n < ω} are all smaller than f in our fixed well-ordering on A which contradicts to the fact that the type of this well-ordering is at most ω.
The Theorem follows easily from Lemma 15. Indeed, pick a maximal wave (W 0 , X 0 ) with respect to D 0 := D where W 0 = {P n } n<ω . Apply Lemma 15 with P 0 ∈ W 0 . The resulting arborescence A 0 contain a unique s → t path P * 0 which is necessarily a forward-continuation of P 0 (usage of a new edge from in D (X 0 ) would lead to dependence). Then by Lemma 15 we have a maximal wave (W 1 , X 1 ) (where X 1 ⊆ X 0 ) with respect to D 1 := D 0 (A(A 0 )) such that W 1 = {P 1 n } 1≤n<ω where P 1 n is a forward continuation of P n . Then we apply Lemma 15 with the D 1 -wave (W 1 , X 1 ) and P 1 1 ∈ W 1 and continue the process recursively. By the construction n<m A(P * n ) is independent for each m < ω. Since M is finitary n<∞ A(P * n ) is independent as well thus P := {P * n } n<ω is a desired paths-system that satisfies the complementarity conditions with X 0 .