On the Infinite Lucchesi-Younger Conjecture I

A dicut in a directed graph is a cut for which all of its edges are directed to a common side of the cut. A famous theorem of Lucchesi and Younger states that in every finite digraph the least size of an edge set meeting every dicut equals the maximum number of disjoint dicuts in that digraph. In this first paper out of a series of two papers, we conjecture a version of this theorem using a more structural description of this min-max property for finite dicuts in infinite digraphs. We show that this conjecture can be reduced to countable digraphs where the underlying undirected graph is $2$-connected, and we prove several special cases of the conjecture.


§1. Introduction
In finite structural graph theory there are a lot of theorems which illustrate the dual nature of certain objects by relating the maximum number of disjoint objects of a certain type in a graph with the minimal size of an object of another type in that graph. More precisely, the size of the latter object trivially bounds the number of disjoint objects of the first type existing in the graph.
This duality aspect of such packing and covering results is closely related to the duality of linear programs appearing in combinatorial optimisation. However, a purely graph theoretic interpretation requires integral solutions of both linear programs, which are hard to detect, if they even exist.
Probably the most well-known example of such a min-max result is Menger's theorem for finite undirected graphs. It states that for any two vertex sets A, B in a finite graph the maximum number of disjoint paths between A and B equals the minimum size of a vertex set separating A from B. In fact, there is a structural reformulation of this quantitative description of this dual nature of connectivity: for any two vertex sets A, B in a finite graph there exists a set of disjoint paths between A and B together with a vertex set separating A from B that consists of precisely one vertex from each of the paths.
While for finite graphs this is an easy corollary from the quantitative version, in infinite graphs it turns out that such a structural version is much more meaningful. While Erdős observed that a version of Menger's theorem based on the equality of infinite cardinals 2 J. PASCAL GOLLIN AND KARL HEUER is quite trivial, he conjectured that the analogue of the structural version is the better way to interpret this dual nature of connectivity. Such a version has been established by Aharoni and Berger [2]. Their theorem restored many of the uses of connectivity duality that the trivial cardinality version could not provide, and hence it influenced much of the development of infinite connectivity theory and matching theory.
Another such min-max theorem was established by Lucchesi and Younger [10] for directed graphs. To state that theorem we have to give some definitions first.
In a weakly connected directed graph D we call a cut of D directed, or a dicut of D, if all of its edges have their head in a common side of the cut. We call a set of edges a dijoin of D if it meets every non-empty dicut of D. Now we can state the mentioned theorem. (ii) F Ď EpDq is a dijoin of D.
(iv) |F X B| " 1 for every B P B.
In this paper we consider the question whether Theorem 1.2 extends to infinite digraphs.
Let us first show that a direct extension of this formulation to arbitrary infinite digraphs fails. To do this we define a double ray to be an undirected two-way infinite path. Now consider the digraph depicted in Figure 1.1. Its underlying graph is the Cartesian product of a double ray with an edge. Then we consistently orient all edges corresponding to one copy of the double ray in one direction and all edges of the other copy in the different direction. Finally, we direct all remaining edges such that they have their tail in the same copy of the double ray. This digraph contains no finite dicut, but it does contain infinite ones. Note that every dicut of this digraph contains at most one horizontal edge, which corresponds to an oriented one of some copy of the double ray, and all vertical edges to the left of some vertical edge. Hence, we cannot even find two disjoint dicuts. However, 3 a dijoin of this digraph cannot be finite, as we can easily find a dicut avoiding any finite set of edges by considering a horizontal edge to the left of the finite set. So we obtain that each dijoin hits every dicut infinitely often in this digraph. Therefore, neither the statement of Theorem 1.2 nor the statement of Theorem 1.1 remain true if we consider arbitrary dicuts in infinite digraphs. Another counterexample for these naive extensions is the countably infinite transitive tournament without a sink, i.e. an orientation of the countably infinite clique without any directed cycles or sinks. We leave the verification of this fact to the reader.
In order to overcome the problem of this example let us again consider the situation in Menger's theorem. There, even in the infinite version, we are only considering finite paths for those objects that we want to pack. Together with the example in Figure 1  Hence, we also make the following conjecture.

Conjecture 1.5.
There exists a nested optimal pair for every weakly connected digraph.
In weakly connected infinite digraphs there are indications that, in contrast to the finite case, Conjecture 1.5 may be strictly stronger than Conjecture 1.3. In Section 3 we will illustrate examples of digraphs with a finitary dijoin which is part of an optimal pair, but not of any nested one.
One of the main results of this paper is the reduction of Conjectures 1.3 and 1.5 to countable digraphs with a certain separability property and whose underlying multigraphs are 2-connected. We call a digraph D finitely diseparable if for any two vertices v, w P V pDq there is a finite dicut of D such that v and w lie in different sides of that finite dicut. (iv) Every edge of D lies in only finitely many finite dibonds of D.
(v) D has no infinite dibond.
The structure of this paper is as follows. In Section 2 we introduce our needed notation and prove some basic tools that we will need throughout the paper. In Section 3 we will This proof is based on the ideas of the negative circuit method developed for more general submodular frameworks by Fujishige [6] and Zimmermann [11]. Instead of just starting with a dijoin of minimum size, the idea of Frank's proof is to start with any dijoin and algorithmically "improve" it with the help of cycles of negative cost in an auxiliary digraph whose definition depends on the dijoin. Once the dijoin can no longer be "improved" some structural properties of the auxiliary graph help in fining the desired set of dibonds which together with the dijoin form a nested optimal pair. §2. Basic notions and tools For basic facts about finite and infinite graphs we refer the reader to [4]. Several proofs, especially in Section 5, base on certain compactness arguments using the compactness principle in combinatorics. We omit stating it here but refer to [4,Appendix A]. Especially for facts about directed graphs we refer to [3].
In general, we allow our digraphs to have parallel edges, but no loops unless we explicitly mention them. Similarly, all undirected multigraphs we consider do not have loops if nothing else is explicitly stated.
Throughout this section let D denote a digraph with vertex set V pDq and edge set EpDq.
We view the edges of D as ordered pairs pu, vq of vertices u, v P V pDq and shall write uv instead of pu, vq, although this might not uniquely determine an edge. In parts where a finer distinction becomes important we shall clarify the situation. For an edge uv P EpDq we furthermore call the vertex u as the tail of uv and v as the head of uv. We denote the underlying undirected multigraph of D by UnpDq.
In an undirected non-trivial path we call the vertices incident with just one edge the endvertices of that path. For the trivial path consisting just of one vertex, we call that vertex also an endvertex of that path. If P is an undirected path with endvertices v and w, we call P a v-w path. For a path P containing two vertices x, y P V pP q we write xP u for the x-u subpath contained in P . Should P additionally be a directed path where v has out-degree 1, then we call P a directed v-w path. We also allow to call the trivial path with endvertex v a directed v-v path. For two vertex sets A, B Ď V pDq we call an undirected path P Ď D an A-B path if P is an a-b path for some a P A and b P B but is disjoint from A Y B except from its endvertices. Similarly, we call an directed path that is 2.1. Cuts and dicuts. Throughout this subsection let D denote a weakly connected digraph. For two vertex sets X, Y Ď V pDq we define E D pX, Y q Ď EpDq as the set of those edges that have their head in X Y and their tail in Y X, or their head in Y X and their tail in X Y . Furthermore, we define We will usually omit the subscript if the graph we are talking about is clear from the context.
If X Y Y " V pDq and X X Y " ∅, we call EpX, Y q a cut of D and refer to X and Y as the sides of the cut. Moreover, by writing EpM, N q and calling it a cut of D we implicitly assume M and N to be the sides of that cut, and by calling an edge set B a cut we implicitly assume that B is of the form EpM, N q for suitable sets M and N .
We call two cuts EpX 1 , Y 1 q and EpX 2 , Moreover, we call a set or sequence of cuts of D nested if its elements are pairwise nested. If two cuts of D are not nested, we call them crossing (or say that they cross).
A cut is said to separate two vertices v, w P V if v and w lie on different sides of that cut.
A minimal non-empty cut is called a bond. Note that a cut EpX, Y q is a bond, if and only if the induced subdigraphs DrXs and DrY s are weakly connected digraphs.
We call a cut EpX, Y q directed, or briefly a dicut, if all edges of EpX, Y q have their head in one common side of the cut. A bond that is also a dicut is called a dibond.
We call D finitely separable if for any two different vertices v, w P V there exists a finite cut of D such that v and w are separated by that cut. Note that if two vertices are separated by some finite cut, then they are separated by some finite bond as well. If furthermore any two different vertices v, w P V pDq can even be separated by a finite dicut, or equivalently a finite dibond, of D, we call D finitely diseparable.
For a vertex set X Ď V pDq we define ‚ δDpXq :" Ý Ñ E pV pDq X, Xq, the set of in-going edges of X; ‚ δDpXq :" Ý Ñ E pX, V pDq Xq, the set of out-going edges of X; ‚ δ D pXq :" δ´pXq Y δ`pXq, the set of incident edges of X.
As before, we will usually omit the subscript if the graph we are talking about is clear from the context. 7 Given a dicut B " Ý Ñ E pX, Y q we call Y the in-shore of B and X the out-shore of B. We shall also write inpBq for the in-shore of the dicut B and outpBq for the out-shore of B.
For undirected multigraphs cuts, bonds, sides, the notion of being nested and the notion of separating two vertices are analogously defined. Hence, we call an undirected multigraph finitely separable if any two vertices can be separated by a finite cut of the multigraph.
Furthermore, in an undirected multigraph G with X, Y Ď V pGq we write EpX, Y q for the set of those edges of G that have one endvertex in X Y and the other in Y X.
(1) Ź B is either empty, or a dicut of D.
(2) Ž B is either empty, or dicut of D.

Note that
Furthermore, note that if B 1 and B 2 are dibonds then B 1^B2 does not need to be a dibond, even if it is non-empty.
A simple double-counting argument yields the following remark.

Remark 2.2.
Let B 1 and B 2 be dicuts of D, and let F Ď EpDq. Then Moreover, if B 1 and B 2 are disjoint, then B 1^B2 and B 1 _ B 2 are disjoint as well.
Let B be a dicut. We call a set B "  We call a tuple pF, Bq a B-optimal pair for D if We call a B-optimal pair pF, Bq for D nested if the elements of B are pairwise nested.
Note that the (nested) optimal pairs as defined in the introduction are precisely the (nested) B fin -optimal pairs.
Using the introduced notation we state the following question, which is the general main topic of our studies in this paper and the second paper [8] of this series.

Question 2.4. For which weakly connected digraphs and classes B of finite dibonds is there a (nested) B-optimal pair?
Note that this question is more general and flexible than Conjecture 1.3 or Conjecture 1.5 but encompasses them by setting B " B fin .
Let B be a class of dicuts of D.
Note that pB ' q ' " B ' , and that by Remark 2.3, if B is finite-corner-closed, then so is B ' .
Throughout this paper we will mostly consider classes of finite dibonds of D which are finite-corner-closed, for example B fin , the class of finite dibonds of D. 9 2.3. Stars and combs. We call an undirected graph a star if it is isomorphic to the complete bipartite graph K 1,κ for some cardinal κ, where the vertices of degree 1 are its leaves and the vertex of degree κ is its centre.
We define a ray to be an undirected one-way infinite path. Any subgraph of a ray R that is itself a ray is called a tail of R. The unique vertex of R of degree 1 is the start vertex of R.
An undirected multigraph that does not contain a ray is called rayless.
A comb C is an undirected graph that is the union of a ray R together with infinitely many disjoint undirected finite paths each of which has precisely one vertex in common with R, which has to be an endvertex of that path. The ray R is called the spine of C.
The endvertices of the finite paths that are not on R together with the endvertices of the trivial paths are the teeth of C.
The following lemma is a basic tool in infinite graph theory. A basic version of this set U 1 Ď U with |U 1 | " |U | such that G either contains a comb whose set of teeth is U 1 or a subdivided star whose set of leaves is U 1 .

Finitely separable multigraphs.
In this section we prove certain size related properties of finitely separable multigraphs using Lemma 2.5.
For a multigraph G we call a subgraph X Ď G a 2-block of G if X is a maximal connected subgraph without a cutvertex. Hence a 2-block of a connected multigraph either consists of a set of pairwise parallel edges in G or is a maximal 2-connected subgraph of G. In a digraph D we call a subdigraph X a 2-block of D if UnpXq is a 2-block of UnpDq.
Let X denote the set of all 2-blocks in G, and C the set of all cutvertices in G. Then the bipartite graph with vertex set X Y C with edge set tcX | c P C, X P X , c P Xu is a tree, the 2-block-cutvertex-tree.
We immediately get the following remark.
Remark 2.6. Let G be a multigraph or a digraph.
(i) Every bond of G is contained in a unique 2-block.
(ii) Bonds of G that are contained in different 2-blocks are nested.

Lemma 2.7.
(i) Every 2-block of a finitely separable multigraph or digraph is countable.
(ii) Every 2-block of a finitely separable rayless multigraph or digraph is finite.
Proof. Let G be a finitely separable multigraph and let X be a 2-block of G. Assume for a contradiction that either X is infinite and rayless, or X is uncountable. Let U be a subset of V pXq with |U | " mint|X|, ℵ 1 u. Applying Lemma 2.5 to U in X, we obtain a subdivided star S 1 in X whose set of leaves L 1 satisfies |L 1 | " |U |. Let c 1 be the centre of S 1 . Using that X is 2-connected, we now apply Lemma 2.5 to L 1 in G´c 1 , which is still connected. Hence, we obtain a subdivided star S 2 in G´c 1 whose set of leaves L 2 satisfies |L 2 | " |L 1 | and L 2 Ď L 1 . Let c 2 denote the centre of S 2 . Now we get a contradiction to G being finitely separable because S 1 and S 2 have infinitely many common leaves in L 2 .
So GrV pS 1 q Y V pS 2 qs contains infinitely many internally disjoint c 1 -c 2 paths, witnessing that c 1 and c 2 cannot be separated by a finite cut of G.
To complete the proof we still need to consider for a contradiction a 2-block X of G whose vertex set is countable (in case (i)) or finite (in case (ii)) but whose edge set is uncountable (in case (i)) or infinite (in case (i)). A contradiction to the fact that X is finitely separable arises by an easy application of the pigeonhole principle to the two-element subsets of V pXq.
Together with Remark 2.6 we obtain the following immediate corollary. We state the following basic lemma without proof. defines an equivalence relation. For v P V pGq we shall write rvs " B for the equivalence class with respect to " B containing v.
Let G{" B denote the digraph, or multigraph respectively, which is obtained from G by identifying the vertices in the same equivalence class of " B and deleting loops. Furthermore, letX :" trxs " B | x P Xu for every set X Ď V pDq, as well asX :" ty P x | x P Xu for every set X Ď V pGq{ " B .
Proposition 2.10. Let G be a digraph or a multigraph and let B be a set of cuts of G.
Then the following statements hold.
(iii) Every cut (or dicut, respectively) EpX, Y q of G{" B is also a cut (or dicut, respectively) of G, and EpX, Y q " EpX,Ỹ q.
(iv) Two cuts in B are nested as cuts of G if and only if they are nested as cuts of G{" B . ( Proof. For the sake of readability we will phrase the proof just for cuts and bonds. The arguments for dicuts and dibonds are analogous. Note that if GrXs is (weakly) connected for some X Ď V pGq, then G{ " B rXs is (weakly) connected as well. Hence statement (i) is immediate.
If EpX, Y q P B, then for every x P X all vertices in rxs " B are contained in X by definition of " B . Analogously, all vertices in rys " B lie in Y for each y P Y . Hence, EpX,Ŷ q " EpX, Y q and is a cut of D{" B . If EpX, Y q is a bond of G, then so it is as a bond of G{ " B by the observation on connectivity of the sides from above. This proves statement (ii).
For statement (iii) let EpX, Y q be a cut of G{" B . By definition of " B we obtain that EpX, Y q is a cut of D as well as M "X and N "Ỹ yielding EpX, Y q " EpX,Ỹ q.
In order to show statement (v), let rvs " B and rws " B be two different vertices of V pG{" B q.
Since v and w are not contained in the same equivalence class, there must exist a cut EpX, Y q P B separating them. By statement (ii) we get that EpX,Ỹ q is a cut of G{" B and it separates rvs " B from rws " B by definition of " B .
We will apply this proposition mostly with the set of all finite bonds of a multigraph G, or the set B fin of all finite dibonds of a digraph D, yielding a multigraph which is finitely separable or a digraph which is finitely diseparable.
Let D be any digraph and let B fin be the set of finite dibonds of D. For ease of notation let " denote the relation " B fin .
Next we characterise the relation v " w for any two vertices v, w. An edge set W is a witness for v " w, if it meets every finite cut that separates v and w in both directions, Hence the existence of a witness for v " w is an obvious obstruction. The whole edge set is similar trivially a witness for v " w. Note that there exists always an inclusion-minimal witness for v " w by Zorn's Lemma.
The following lemmas tell us that given a minimal witness W for v " w, all vertices incident with an edge of W are also equivalent to v with respect to ".

Lemma 2.11.
Let v " w for two vertices v, w P V pDq. Then a minimal witness W for v " w also witnesses v " y for any y P V pDrW sq.
Proof. Let W be a minimal witness for v " w. Now suppose for a contradiction that there is a y P V pDrW sq which is separated from v by a finite dibond B " both vertices v and w have to lie on the same side of B, namely X. We claim that W 1 :" W X EpDrXsq also witnesses v " w. This would be a contradiction to the minimality of W as y is incident with an edge of W both of whose endvertices lie in Y since W X B " ∅.
Let EpM, N q be a finite cut of D separating v and w, say with v P M and w P N .
Since EpX X M, Y Y N q is also a finite cut, but W X EpX X M, Y q " ∅, we obtain Corollary 2.12. Let v " w for two vertices v, w P V pDq. Then a minimal witness W for v " w induces a strongly connected digraph DrW s.
Proof. Assume for a contradiction that there is a dicut Ý Ñ E pX, Y q separating some vertices w 1 , w 2 P W . By Lemma 2.11, W is also a witness for w 1 " w 2 , contradicting Let us close this subsection with the following corollary of Proposition 2.10 and Lemma 2.7(i). Moreover, we denote the relation " Bf n by «.

13
Note that since v " w implies that v « w for all v, w P V pF q, we obtain that " induces an equivalence relation on V pD{«q. Since moreover the set of finite dibonds of D{« equals the set of finite dibonds of D by Proposition 2.10, we obtain the following remark.
Remark 2.14. pD{«q{" " D{" The aim of this subsection is to show that if D is rayless, then so is D{". The analogous statement for the relation « is proven by an easy construction.

Remark 2.15.
If D is rayless, then D{« is rayless as well.
Proof. Suppose for a contradiction that D is rayless but R " rv 0 s « rv 1 s « . . . is a ray in D{«. For each i P N let v 1 i P rv i s « and v 2 i`1 P rv i`1 s « be the endvertices of the edge rv i srv i`1 s P EpRq seen in D. To arrive at a contradiction, we will construct a ray in D inductively. Let P 0 be the trivial path containing just v 1 0 . Assume for i ą 0 that there is a j ě i such that that P i is a rv 0 s « -rv j s « -path which contains P i´1 and is internally disjoint If P is disjoint from rv k s « for all k ą j, then let P i`1 be concatenation of the paths P i , P and the edge v 2 j v 1 j`1 . Otherwise let w be the first vertex of P in rv k s « for some k ą j and let P i`1 be the concatenation of P i with v 3 j P w. In both cases P i`1 satisfies the desired properties and Ť iPN P i is the desired ray in D.
Before we can prove the analogue for digraphs, we have to prepare some lemmas. The first is about inclusion-minimal edge sets witnessing the equivalence of two vertices with respect to " in digraphs whose underlying multigraph is rayless. Proof. By Remarks 2.14 and 2.15 we may assume without loss of generality that D is finitely separable. Suppose for a contradiction that D is rayless but R " rv 0 s " rv 1 s " . . . is a ray in D{". For each i P N let v 1 i P rv i s " and v 2 i`1 P rv i`1 s " be the endvertices of the edge rv i s " rv i`1 s " P EpRq seen in D. Furthermore, let W i be an inclusion-minimal witness for v 2 i " v 1 i`1 for every i P N with i ě 1. We know by Lemma 2.11 that each W i is completely contained in rv i s " . By Corollary 2.12 and Lemma 2.16 each W i is strongly connected and finite. Since each W i is completely contained in rv i s " , we get that W i X W j " ∅ holds There exist finitary dijoins that are part of an optimal pair, but of no nested optimal pair. This is severely different from finite digraphs. There, we could always keep the dijoin F of any optimal pair pF, Bq and just iteratively 'uncross' all dicuts of B, yielding a set B 1 of nested disjoint dicuts such that pF, B 1 q is a nested optimal pair. We illustrate this uncrossing process in the proof of Lemma 5.1.
Let us now describe the first example. Before we analyse D 1 in detail, let us define D 1 properly.
Let A " ta i | i P Nu and B " tb i | i P Nu be two disjoint countably infinite sets. Additionally, let r be some set which is neither contained in A nor in B. Now we set Next we define the sets E 1 :" We complete the definition of D 1 by setting Next consider the set E 2 of grey edges in the left instance of D 1 depicted in Figure 3 can easily find a nested optimal pair for D 1 in which F L features. Hence, D 1 is not a counterexample to Conjecture 1.5.
In the right instance of D 1 depicted in Figure 3.1, the set of grey edges E 1 Y tb 0 ru, call it F R , also forms a finitary dijoin. And again we can easily find an optimal pair for D 1 in which F R features. However, no matter which finite dicut we choose which contains the grey edge adjacent to r, it cannot be nested with all the finite dicuts we choose for all the other edges of F R . Therefore, F R does not feature in any nested optimal pair for D 1 .
Let us now consider another example, witnessing the same behaviour of finitary dijoins as Example 1 does. However, the structure of the digraph D 2 in the following example is rather different from D 1 . In particular, D 2 is a locally finite digraph, i.e. every vertex is incident with only finitely many edges.

Example 2.
Consider the infinite weakly connected digraph D 2 depicted in Figure 3.2.
We first define vertex set of D 2 as V pD 2 q :" ) .
Note that for each px, yq P V pD 2 q both px, y`1q and px´1, yq are in V pD 2 q as well. We define E 1 :" tpx, y`1qpx, yq | px, yq P V pD 2 qu and E 2 :" tpx´1, yqpx, yq | px, yq P V pD 2 qu.
Finally, we define the edge set of D 2 by Now consider the set of dashed grey edges in Figure 3.2, ) .
It is an easy exercise to check that F d forms a finitary dijoin of D 2 which also features in a nested optimal pair for D 2 . Therefore, the digraph D 2 is also no counterexample to Conjecture 1.5.
In contrast to this, let us now consider the set of uninterruptedly grey edges in Figure 3.2, Again it is easy to check that F s forms a finitary dijoin of D 2 . However, F s is not part of any nested optimal pair for D 2 . This is not difficult to prove using the fact that F d is a finitary dijoin of D 2 as well. We leave this proof to the reader. §4. Reductions for the Infinite Lucchesi-Younger Conjecture In this section we prove some reductions for Conjecture 1.3 and Conjecture 1.5 in the sense that it suffices to solve these conjectures on a smaller class of digraphs. We begin by reducing these conjectures to finitely diseparable digraphs via the following lemma.

Then pF, Bq is a (nested) B-optimal pair for D if and only if it is a (nested) B-optimal
pair for D{ " B .
Proof. Note first that by Proposition 2.10 D{" B is weakly connected and that B is also a set of dibonds of D{" B .
Suppose pF, Bq is a (nested) B-optimal pair for D. Then F is still a subset of EpD{" B q since each edge of F lies on some dibond B P B Ď B. Hence, F is still a B-dijoin of D{" B , and pF, Bq is indeed a (nested) B-optimal pair for D{" B , again by Proposition 2.10.
Similarly, if pF, Bq is a (nested) B-optimal pair for D{ " B , then so it is for D, again by Proposition 2.10.
The next reduction of Conjecture 1.3 and Conjecture 1.5 tells us that we can restrict our attention also to digraphs whose underlying multigraph is 2-connected. (i) For each X P X the set B X :" tB P B | B Ď EpXqu is a class of dibonds of X and B " 9 Bq is a (nested) B-optimal pair for D, then pF X , B X q is a (nested) B X -optimal pair for every X P X , where F X :" F X EpXq and B X :" tB P B | B Ď EpXqu.
(iii) If pF X , B X q is a (nested) B X -optimal pair for every X P X , then pF, Bq is a (nested) B-optimal pair for D, where F :" Ť XPX F X and B :" Proof. Let X be a 2-block of D. By Remark 2.6 every dibond B P B is either contained in EpXq and hence a dibond of X, or disjoint to EpXq. Vice versa, every dibond of X is a dibond of D as well. Statement (i) is now easy to check.
For statement (ii), let X P X and let pF, Bq be a (nested) B-optimal pair for D. Then by just translating the definitions we obtain that pF X EpXq, B X q is a (nested) B X -optimal pair for D, as well as for X.
Now we show that statement (iii) is true. So let us assume that pF X , B X q is a (nested) B X -optimal pair for every X P X . With statement (i) (and Remark 2.6(ii)) we immediately get that with pF, Bq is a (nested) B-optimal pair for D.
We can now close this section by proving Theorem 1. 6 Proof. Let D be any weakly connected digraph and let B fin the set of finite dibonds of D. We know by Proposition 2.10 that D{" B fin is a weakly connected and finitely diseparable digraph, and so is every 2-block of it. Furthermore, Corollary 2.13 yields that each 2-block of D{" B fin is countable. By our assumption we know that Conjecture 1.3 (or Conjecture 1.5, respectively) holds for every countable 2-block of D{" B fin . So using Lemma 4.2 we obtain a (nested) optimal pair for D{" B fin . Then we also obtain an optimal pair for D by Lemma 4.1.

§5. Special cases
In this section we prove some special cases of Conjecture 1.5, or more precisely cases of Proof. We start by proving the implication from (i) to (ii). Let F be a B-dijoin of D of finite size. Then, by definition, we can find at most |F | many disjoint dibonds in B.
For the implication (ii) to (i) note that for any inclusion-wise maximal set B of disjoint dibonds in B the set F :" Ť B is a finite B-dijoin of D.
The implication from (ii) to (iii) is immediate, even if B is not finite-corner-closed.
Finally, we assume statement (iii) and that B is finite-corner-closed, and we prove statement (i).
Suppose that for some finite set B Ď B of pairwise disjoint and pairwise nested finite dibonds which is of maximum size there is some dibond A P B which is disjoint to each dibond in B. Without loss of generality, let B and A be chosen such that the number of dibonds in B that cross A is of minimum size among all possible choices.
Let B P B be chosen such that A and B cross and either inpBq (first case) or outpBq (second case) is inclusion-minimal among all sides of the elements of B that cross A.
In the first case we consider the dicut A^B P B ' . Note that since both A and B are dibonds, the out-shore of A^B induces a weakly connected digraph. Hence an easy case analysis shows that any dibond in its decomposition into dibonds in B is nested with every dibond in B as well as with each other. In particular, A^B is a dibond in B, since otherwise it would contradict the maximality of B. Moreover, let A 1 be any dibond appearing in the decomposition of A _ B into dibonds in B. As before, we can show that A 1 is nested with A^B, as well as with any dibond in B which is nested with A. And since B 1 :" pB tBuq Y tA^Bu is a set of pairwise disjoint dibonds in B and A 1 crosses strictly fewer dicuts in B 1 than A crosses in B, the pair B 1 and A 1 contradicts the choice of B and A. In the second case the same argument works with the roles of A^B and In any case, this contradicts the existence of such a set B and such a dibond A. Therefore, We claim that pF, Bq is a (nested) B-optimal pair for D. We already know by definition that B is a (nested) set of disjoint finite dicuts in B and that F Ď Ť B with |F X B| " 1 for every B P B. It remains to check that F is a B-dijoin of D. So let B 1 P B. Then the set N 1 :" B 1 Y Ť B is also finite and B 1 is a finite dicut of D.N 1 . Since pF, Bq is also a nested optimal pair for D.N 1 , we know that F X B 1 ‰ ∅ holds, which proves that F is a B-dijoin of D.

Every edge lies in only finitely many dibonds and reductions to this case.
We continue with verifying another special case of Question 2.4. The proof is also based on a compactness argument. However, we need to choose the set up for the argument more carefully. (1) F B intersects every element of B, We claim that pF D , B D q is a nested optimal pair for D. First we verify that F D is a finitary dijoin of D. Let B be any finite dibond of D. Then F D meets B, because pF D X B, B D X y tBuq is a nested pre-optimal pair for D.B. So F D is a finitary dijoin of D.
Next consider any element e P F D . By definition of F D we know that e P B e holds for some finite dibond B e of D. Using again that pF D X B e , B D X z tB e uq is a nested pre-optimal pair for D.B e , we get that e P Ť B D . So the inclusion F D Ď Ť B D is valid.
Given any B D P B D we know that pF D X B D , B D X z tB D uq is a nested pre-optimal pair for D.B D . Hence, |F D X B| " 1 holds for every B P B D X z tB D u. Especially, |F D X B D | " 1 is true because B D P B D X z tB D u. Finally, let us consider two arbitrary but different elements B 1 and B 2 of B D . We know that pF D X pB 1 Y B 2 q, B D X { tB 1 , B 2 uq is a nested pre-optimal pair for D.pB 1 Y B 2 q. Therefore, B 1 and B 2 are disjoint and nested. This shows that pF D , B D q is a nested optimal pair for D and completes the proof of this lemma.
The next lemma can be used together with Lemma 5.3 to deduce that Conjecture 1.5 holds for weakly connected digraphs without infinite dibonds. Proof. We construct with a compactness argument a dibond containing e ": vw that is distinct from every finite dibond. As noted before, we obtain the following corollary. (1) M is a matching of H.
(2) A is a cover of H.
(4) |M X A| " 1 for every M P M. 23 We call such a pair pM, Aq an optimal pair for H. Now we are able to state the original problem on infinite hypergraphs posted by Aharoni. We shall now point out that, in full generality, this problem has a negative answer by stating a certain infinite hypergraph H. For this, consider the digraph in Figure 1.1, call it D. Let B denote the set of all dicuts of D. We now define the hypergraph H " pV, Eq by setting V " EpDq and E " B. As discussed in the introduction, H does not have the Kőnig property, since we cannot even find two disjoint hyperedges, but we need infinitely many vertices of V to cover all hyperedges. However, for every non-empty finite subset F of E we can find one vertex of V covering all hyperedges of pV, F q.
As noticed in the introduction, H does not have any finite hyperedges. This motivates us to modify Problem 6.1 to include only hypergraphs of finite character.