Every infinitely edge-connected graph contains the Farey graph or $T_{\aleph_0}\!\ast t$ as a minor

We show that every infinitely edge-connected graph contains the Farey graph or $T_{\aleph_0}\!\ast t$ as a minor. These two graphs are unique with this property up to minor-equivalence.

. The Farey graph Figure 2. The graph T ℵ0 * t

Introduction
The Farey graph, shown in Figure 1 and surveyed in [12,18,19], plays a role in a number of mathematical fields ranging from group theory and number theory to geometry and dynamics [6]. Curiously, graph theory is not among these. In this paper we show that the Farey graph plays a central role in graph theory too: it is one of two infinitely edge-connected graphs that must occur as a minor in every infinitely edge-connected graph. Previously it was not known that there was any set of graphs determining infinite edge-connectivity by forming a minor-minimal list in this way, let alone a finite set. Ramsey theory and the study of connectivity intersect in the problem of finding for any given connectivity k a small set of k-connected subgraphs that occur in every k-connected graph, and thereby characterise k-connectedness. To keep these unavoidable sets small for k ≥ 3, the subgraph relation referred to above is usually relaxed to the graph minor relation. Here, a graph is a minor of a graph G if it can be obtained from a subgraph of G by contracting connected (possibly infinite) induced disjoint subgraphs [7]. We refer to [7, §9.4] or the introduction of [10] for surveys on the known results for this problem and its variations [7, 9-11, 13, 15]. Such sets of minor-minimal k-connected graphs are known only for k ≤ 4, and only for finite graphs [15]. These results of Oporowski, Oxley and Thomas were generalised to k > 4 by Geelen and Joeris [9] for finite graphs, and by Gollin and Heuer [10] for infinite graphs, but with a different notion of connectivity.
For infinite connectivity, the problem asks for a small selection of infinitely connected graphs such that every infinitely connected graph contains at least one of the selected graphs as a minor. Here, 'infinitely connected' can be understood in two ways. When it is understood as 'infinitely vertex-connected', the answer is already known: Every infinitely connected graph contains the countably infinite complete graph K ℵ0 as a minor [7, §8.1]. But when 'infinitely connected' is understood as 'infinitely edge-connected' then, as we shall see, K ℵ0 is not the answer, and in fact no answer has been known. Indeed it is not even clear a priori that there exists a finite set of unavoidable infinitely edge-connected minors. Any such unavoidable infinitely edge-connected minors will be countable, because in every infinitely edge-connected graph we can greedily find a countable infinitely edge-connected subgraph. But the countable graphs are not known to be well-quasi-ordered by the minor-relation. It is therefore not clear that any minor-minimal set of infinitely edge-connected graphs must be finite, nor even that such a minimal set exists.
In this paper we find a pair of infinitely edge-connected graphs that occur unavoidably as minors in any infinitely edge-connected graph, and which are unique with this property up to minor-equivalence: the Farey graph F , and the graph T ℵ0 * t obtained from the infinitely-branching tree T ℵ0 by joining an additional vertex t to all its vertices ( Figure 2). Theorem 1. Every infinitely edge-connected graph contains either the Farey graph or T ℵ0 * t as a minor.
The uniqueness of the pair {F, T ℵ0 * t }, up to minor-equivalence, follows from the fact that they are not minors of each other (Lemmas 3.1 and 3.2): Theorem 2. Let M be any set of infinitely edge-connected graphs such that every infinitely edge-connected graph has a minor in M and no element of M is a minor of another. Then M consists of two graphs, of which one is minor-equivalent to the Farey graph and the other is minor-equivalent to T ℵ0 * t.
Theorem 1 is best possible also in the sense that one cannot replace 'minor' with 'topological minor' in its wording (Theorem 3.4).
Since both the Farey graph and T ℵ0 * t are planar, our result implies that every infinitely edge-connected graph contains a planar infinitely edge-connected graph as a minor. Thus, in this sense, infinite edge-connectivity is an inherently planar property.
This paper is organised as follows. Section 2 formally introduces the Farey graph. In Section 3 we show that the Farey graph and T ℵ0 * t are not minors of each other, and deduce Theorem 2. Theorem 3.4 above is proved there as well. We outline the overall strategy of the proof of Theorem 1 in Section 4. The proof itself consists of two halves. The first half of the proof is carried out in Section 5, and the second half is carried out in Section 6.
Acknowledgement. I am grateful to Konstantinos Stavropoulos for stimulating conversations.

Preliminaries
We use the notation of Diestel's book [7] and assume familiarity with the basics of separation systems [8].
Two graphs are said to be minor-equivalent if they are minors of each other. Note that minor-equivalence defines an equivalence relation on every set of graphs.
A halved Farey graph of order 0 is a K 2 whose sole edge is coloured blue. Recursively, a halved Farey graph of order n > 0 is an edge-coloured graph G that is obtained from a halved Farey graph H of order n − 1 as follows. The graph G is obtained from H by adding a new vertex v e for every blue edge e of H and joining v e precisely to the endvertices of e. The edge-colouring of G colours the new edges blue whereas all edges of H ⊆ G receive black. Since all halved Farey graphs of order n are isomorphic to each other via isomorphisms that respect their edge-colourings, we speak of the halved Farey graphF n of order n and assume thatF n+1 arises fromF n as described above. The halved Farey graph is the union F := n∈NF n of all halved Farey graphsF n of order n, and it does not come with an edge-colouring.
The Farey graph is constructed quite similarly. A Farey graph of order 0 is a K 2 whose sole edge is coloured blue. A Farey graph of order 1 is obtained from a C 4 with blue edges by adding one black chord. Recursively, a Farey graph of order n > 1 is an edge-coloured graph G that is obtained from a Farey graph H of order n − 1 as follows. The graph G is obtained from H by adding a new vertex v e for every blue edge e of H and joining v e precisely to the endvertices of e. The edge-colouring of G colours the new edges blue whereas all edges of H ⊆ G receive black. Again, we speak of the Farey graph F n of order n. The Farey graph is the union F := n∈N F n of all Farey graphs F n of order n. Like the halved Farey graph, it does not come with an edge-colouring. Note that the Farey graph can be obtained from two copies of the halved Farey graph by pasting them alongF 0 . This recursive construction of the Farey graph is based on [6]. Proof. The halved Farey graph is a subgraph of the Farey graph. Conversely, the Farey graph is a minor of the halved Farey graph: if e is a blue edge ofF 1 , then the Farey graph is the contraction minor ofF − e whose sole non-trivial branch set is V (F 0 ) = K 2 , i.e., (F − e)/V (F 0 ) ∼ = F .

Uniqueness and topological minors
3.1. Uniqueness. In this section we show that the pair {F, T ℵ0 * t } is unique up to minor-equivalence: Let H be any set of infinitely edge-connected graphs such that every infinitely edge-connected graph has a minor in H and no element of H is a minor of another. Then H consists of two graphs, of which one is minor-equivalent to the Farey graph and the other is minor-equivalent to T ℵ0 * t.
This will follow easily from the following two lemmas: Lemma 3.1. The Farey graph is not a minor of T ℵ0 * t.
Proof. The Farey graph contains two disjoint cycles, but T ℵ0 * t does not.
Lemma 3.2. The graph T ℵ0 * t is not a minor of the Farey graph.
Proof of Theorem 2. We write G = {F, T ℵ0 * t } and note that neither element of G is a minor of another by Lemmas 3.1 and 3.2. Every graph H ∈ H contains a graph G ∈ G as a minor (Theorem 1) which in turn contains a graph H ∈ H as a minor, and then H G H implies H = H because no element of H is a minor of another. Thus, every graph in H is minor-equivalent to some graph in G and, conversely, every graph in G is minor-equivalent to some graph in H by symmetry. Since no two graphs in H or in G are comparable with regard to the minor-relation, we deduce that minor-equivalence induces a bijection between H and G.
Showing that T ℵ0 * t is not a minor of the Farey graph requires more effort, and some preparation. A comb is the union of a ray R (the comb's spine) with infinitely many disjoint finite paths, possibly trivial, that have precisely their first vertex on R. The last vertices of those paths are the teeth of this comb. Given a vertex set U , a comb attached to U is a comb with all its teeth in U , and a star attached to U is a subdivided infinite star with all its leaves in U . The following lemma is [7,Lemma 8.2.2], see also the series [2][3][4][5]. Proof of Lemma 3.2. Since K 2,ℵ0 is a subgraph of T ℵ0 * t, it suffices to show that the Farey graph does not contain K 2,ℵ0 as a minor. So let us assume for a contradiction that the Farey graph contains a K 2,ℵ0 minor. By applying the star-comb lemma inside the branch sets of the two infinite-degree vertices of K 2,ℵ0 if necessary, and using that the Farey graph does not contain infinitely many independent paths between any two of its vertices, we find that our model of K 2,ℵ0 contains a subdivision G of one of the following two graphs G 1 and G 2 . The graph G 1 is the ladder with every rung subdivided exactly once, i.e., it is the disjoint union of two rays R = v 1 v 2 . . . and R = v 1 v 2 . . . with infinitely many disjoint R-R paths v n z n v n (n ∈ N). And the graph G 2 is obtained from G 1 by contracting R to a single vertex that we call d.
In either case, the sole end of G ⊆ F is included in a unique end ω of F . The end ω chooses, for every n ∈ N, a blue edge e n ∈ F n with vertex set X n for which it lives in the component C n of F − X n avoiding F n . Then C n has neighbourhood X n , and so does the other component D n of F − X n . We remark that, by the construction of the Farey graph, for every vertex of F there is a number n such that the vertex is not contained in C n . For all n the two vertex sets X n and X n+1 together induce a triangle ∆ n in F . We write x n for the vertex in which X n and X n+1 meet, and we write Y n for vertex set consisting of the other two vertices of the triangle ∆ n . The graph F − ∆ n has precisely three components, namely D n and C n+1 and a third component with neighbourhood Y n which we denote by H n .
First, we consider the case that G ⊆ F is a subdivision of G 1 , and we writeR andR for the subdivisions of R and R in G. Then there cannot be a number N such that x n = x N for all n ≥ N : Indeed, for every k ∈ N there is a number f (k) ≥ k such that both v k and v k are not contained in C f (k) and, as a consequence, x f (k) must be contained in v kR ∪ v kR . Thus, every vertex of F lies in a component D n eventually (and ω is undominated). Let N be the least number for which the first vertices ofR andR lie in D N . To derive a contradiction from G ⊆ F , let us consider anyR-R path P ⊆ G that lies entirely in the component C N , and consider the maximal number n for which P avoids D n , noting n ≥ N . Since the two raysR andR induce a bipartition of the 2-set X n+1 , the path P cannot meet C n+1 without contradicting the maximality of n. Therefore, the path P is contained entirely in F [H n ∆ n ]. Without loss of generality we have x n ∈R. Then Y n ⊆R follows. But now the non-empty subpathP must be contained in H n , contradicting that H n has neighbourhood Y n ⊆R .
Second, we consider the case that G ⊆ F is a subdivision of G 2 , and again we writeR for the subdivision of R in G. Since d ∈ G 2 dominates the end of G 2 , the end ω is dominated in F by d. Let N be the least number such that both d and the first vertex ofR are not contained in C N . Then d = x N = x n for all n ≥ N because d dominates ω. Thus, Y n ⊆R for all n ≥ N . Now consider any d-R path P ⊆ G withdP ⊆ C N and choose n maximal with the property that the non-empty subpathP avoids D n , noting n ≥ N . ThenP ⊆ H n follows because of Y n ⊆R, contradicting that d = x n does not lie in the neighbourhood Y n of H n .

3.2.
Minor versus topological minor. Theorem 1 is best possible in the sense that one cannot replace 'minor' with 'topological minor' in its wording: There exists an infinitely edge-connected graph that contains neither the Farey graph nor T ℵ0 * t as a topological minor.
Proof. By a recent result [14] there exists an infinitely edge-connected graph G which does not contain infinitely many edge-disjoint pairwise order-compatible paths between any two of its vertices. Here, two u-v paths are order-compatible if they traverse their common vertices in the same order. Then the graph G does not contain a subdivision of the Farey graph or of T ℵ0 * t because both the Farey graph and T ℵ0 * t have pairs of vertices with infinitely many edge-disjoint pairwise order-compatible paths between them.

Overall proof strategy
Our aim for the remainder of this paper is to show that every infinitely edge-connected graph contains either the Farey graph or T ℵ0 * t as a minor (Theorem 1). The proof consists of two halves. In the first half (Section 5) we show that every infinitely edge-connected graph without a T ℵ0 * t minor is 'robust' (Theorem 5.13), explained below. Then, in the second half (Section 6), we employ Theorem 5.13 to prove that every infinitely edge-connected graph without a T ℵ0 * t minor must contain a Farey graph minor, completing the proof of Theorem 1.
The Farey graph and T ℵ0 * t are both infinitely edge-connected, but in different ways. The infinite edge-connectivity of the Farey graph, on the one hand, is robust in that deleting the two endvertices of an edge always leaves only infinitely edge-connected components. The infinite edge-connectivity of T ℵ0 * t, on the other hand, is fragile in that deleting t results in a tree. In the first half of the proof of Theorem 1 we show that every infinitely edge-connected graph without a T ℵ0 * t minor is essentially robust, not fragile (Theorem 5.13).
In the second half of the proof of Theorem 1 we construct a Farey graph minor in an arbitrary infinitely edge-connected T ℵ0 * t free graph G. By Lemma 2.1 it suffices to construct a halved Farey graph minor. Using that G is robust by Theorem 5.13, we shall essentially prove the following assertion: For every two vertices u and v of G there exist two induced subgraphs H u , H v ⊆ G containing u and v respectively and which satisfy the following conditions: is finite, non-empty and connected in G; (ii) both H u /X and H v /X are infinitely edge-connected; (iii) X avoids u and v; (iv) uX is an edge of H u /X and vX is an edge of H v /X. If we choose u and v to form an edge of G, then the three vertices u, v and X span a triangleF 1 in (H u ∪ H v )/X. And since H u /X and H v /X are both infinitely edgeconnected and robust again, we can reapply the assertion in (H u /X) − uX to u and X, and in (H v /X) − vX to v and X. By iterating this process, we obtain a halved Farey graph minor in the original graph G at the limit, and this will complete the proof of Theorem 1.

Robustness
The aim of this section is to prove Theorem 5.13 which has been outlined in the previous section. Our proof proceeds in three steps. First, we provide some tools that will help us to (i) identify infinitely edge-connected 'parts' of arbitrary graphs and (ii) allow us to distinguish all these 'parts' at once in a tree-like way. In the second step, then we employ these tools to analyse the components of G − u − v for infinitely edge-connected graphs G and vertices u, v of G. In the third step, we proceed to prove Theorem 5.13.

5.1.
Finitely separating spanning trees. Let G be any graph. Two vertices of G are said to be finitely separable in G if there is a finite set of edges of G separating them in G. If every distinct two vertices of G are finitely separable, then G itself is said to be finitely separable. An equivalence relation ∼ = ∼ G is declared on the vertex set of G by letting x ∼ y whenever x and y are not finitely separable. The graphG is defined on V (G)/∼ by declaring XY an edge whenever X = Y and there is an X-Y edge in G. Note that the graphG is always finitely separable. A spanning tree T of G is finitely separating if all its fundamental cuts are finite. The following theorem is Theorem 6.3 in [1] and Theorem 5 in [4].
Theorem 5.1. Every connected finitely separable graph has a finitely separating spanning tree.
Usually, we will employ Theorem 5.1 to find a finitely separating spanning tree T ofG that we will then use to analyse the overall structure of G with regard to infinite edge-connectivity. In this context, the nodes of T ⊆G can also be viewed as the vertex sets of G that they formally are. When we view a node of T as a vertex set of G we will refer to it as part for clarity.
If G is any graph and X ⊆ V (G) is any vertex set, then we denote by ∂X = ∂ G X the subset of X formed by the vertices in X that send an edge in G to V (G) X. If {A, B} ∈ B ℵ0 (G) then { A , B } ∈ B ℵ0 (G): Between every two ∼-classes U and W of G there are only finitely many edges, because u ∈ U is separated from w ∈ W by a finite cut of G and then U and W must respect this finite cut. Hence the finitely many A-B edges inG give rise to only finitely many ( A)-( B) edges in G, and these are all. Using that G contains for all ∼-equivalent vertices x and y an x-y path avoiding the finitely many ( A)-( B) edges, it is straightforward to show that both G[ A ] and G[ B ] are connected.
As a consequence, every finitely separating spanning tree T ofG gives rise to an ) for T 1 and T 2 the two components of T − t 1 t 2 containing t 1 and t 2 respectively. (For the definition of an S-tree see [7, §12.5].) The part of a star can be made connected for a reasonable price: Inductively, we construct an ascending sequence (For whenever a path in G connecting two given vertices in G ∞ uses vertices that are not in G ∞ , then the path crosses one of the bonds F j , and the number of bonds crossed can be decreased by replacing path segments with detours in T ⊇ ∂B j because T ⊆ G ∞ . Therefore, choosing a path that crosses as few bonds F j as possible will suffice to find a path that lies entirely in G ∞ .) To start the construction, let T 0 be any finite tree in G[B i * ] that contains ∂B i * . At step n + 1 of the construction, suppose that we have already constructed T n and J n . As T n is finite, we find an index i n ∈ I J n for which A in avoids T n , ensuring T n ⊆ G n+1 . To ensure that T n can be extended in G n+1 to a finite tree T n+1 that contains ∂B in , it suffices to show that G n+1 is connected. Given any two vertices in G n+1 , consider any path between them in G[B in ], chosen to cross as few of the finite bonds F j with j ∈ J n as possible. Then the path avoids all these F j , for otherwise the number of bonds crossed could be decreased by replacing path segments with detours in T n ⊇ j∈Jn ∂B j . Therefore, G n+1 is connected.

5.2.
Analysing the components. Now we are ready to analyse the components of G − u − v for infinitely edge-connected graphs G and vertices u, v of G. The main results here are the two Lemmas 5.3 and 5.8. Here is the first main lemma: Lemma 5.3. Suppose that G is an infinitely edge-connected graph, that u, v are two distinct vertices of G, and that C is a component of G − u − v. IfC has a finitely separating spanning tree that contains a subdivision of the infinite binary tree, then G[C + u + v] contains T ℵ0 * t as a minor.
Proof. Consider any finitely separating spanning tree ofC that contains a subdivision of the infinite binary tree. Then this spanning tree also contains T ℵ0 as a contraction minor which gives rise to a B ℵ0 (C)-tree (T, α) with T isomorphic to T ℵ0 . Next, we fix any root r ∈ T , and for every edge e ∈ T we fix → e(r) as its orientation pointing away from the root r.
By replacing T with one of its subtrees and restricting α accordingly, we may even assume that O u is cofinal in O. In fact, then O u = O follows as O u is down-closed in O. We will use this to show For this, we enumerate the vertices of T ℵ0 as x 0 , x 1 , . . . such that every x n is neighbour to some earlier x k (k < n). Inductively, we construct a sequence W 0 , W 1 , . . . of disjoint connected vertex sets W n ⊆ V (C), a sequence w 0 , w 1 , . . . of vertices w n ∈ W n , and a sequence t 0 , t 1 , . . . of distinct nodes t n ∈ T such that, for all n ∈ N: Once the construction is completed, the sets W n and {u} will give rise to a model of T ℵ0 * t in G[C + u] by (i) and (ii).
At the construction start, we choose any neighbour w 0 of u in C (which exists as O u = O and T is infinite), guaranteeing (i). Then t 0 is defined by (iii). Applying Lemma 5.2 in C to the infinite star α[ → F t0 ] yields an infinite substar whose connected part W 0 ⊆ V (C) contains w 0 and satisfies both (ii) and (iv) trivially.
At step n > 0 of the construction, consider the k < n for which x k x n is an edge of T ℵ0 , and pick an edge → e ∈ O ∩ ( → F t k ) * that (iv) provides for k ≤ n − 1. If we write α( → e) = (B, A), then the vertex set W k contains ∂ C B while A is avoided by all W i with i ≤ n − 1. Using O u = O we find a neighbour w n of u in A giving (i), and w n defines t n by (iii). Then we apply Lemma 5.2 in C to the infinite star where we take i * ∈ I to be the index of the separation α( → e). This yields an infinite substar whose connected part W n ⊆ V (C) contains w n and satisfies (ii) because W n contains ∂ C A while W k contains ∂ C B. Using the infinite substar and the choice of → e it is straightforward to verify (iv) for all k ≤ n.
Our second main lemma, Lemma 5.8, requires some preparation. An arrow from u to v is a graph G that arises from the two vertices u and v by disjointly adding an infinitely edge-connected graph H, adding a u-H edge uh, and adding infinitely many v-(H − h) edges. Then H is the arrow's payload, u is its nock and v is its head.
An arrow barrage from u to v is a countably infinite union n∈N A n of arrows A n from u to v such that A n and A m do not meet in any vertices other than u and v for all n = m. Then u and v are the nock and head of the arrow barrage.
When we say that some graph contains an arrow (barrage) minor from x to y for two vertices x and y, we mean that the graph contains an arrow (barrage) minor such that the branch set corresponding to the arrow (barrage)'s nock contains x while the branch set corresponding to the arrow (barrage)'s head contains y.
The next definition captures the concept of recursive pruning that Diestel describes in his book [7] as follows: Definition 5.5 (Recursive pruning). Let T be any tree, equipped with a root and the corresponding tree-order on its vertices. We recursively label the vertices of T by ordinals, as follows. Given an ordinal α, assume that we have decided for every β < α which of the vertices of T to label β, and let T α be the subgraph of T induced by the vertices that are still unlabelled. Assign label α to every vertex t of T α whose up-closure t Tα = t T ∩ T α in T α is a chain. The recursion terminates at the first α not used to label any vertex; for this α we put T α =: T * . We call T recursively prunable if every vertex of T gets labelled in this way, i.e., if T * = ∅.

]). A rooted tree is recursively prunable if and only if it contains no subdivision of the infinite binary tree.
The next lemma is an observation that we will use often: Lemma 5.7. Suppose that G is an infinitely edge-connected graph, that u, v are two distinct vertices of G, and that C is a component of G − u − v. If T is a finitely separating spanning tree ofC and t ∈ T has finite degree in T , then C[t] is infinitely edge-connected and either u or v sends infinitely many edges in G to the part t ⊆ V (C).
Proof. As t has finite degree in T , the finite fundamental cuts of the edges of T incident with t together give rise to a finite cut of C with the part t as one of its sides. Thus, in the graph G every vertex in the part t can send only finitely many edges to C − t, at most one edge to each of u and v, and some edges to the rest of the part t. As every vertex of the infinitely edge-connected graph G has infinite degree, it follows that the part t must be infinite. And since no two vertices in t are finitely separable in C while t is separated from the rest of C by a single finite cut, it follows that C[t] is infinitely edge-connected. Finally, at least one of u and v sends infinitely many edges to the part t, because otherwise t is separated from the rest of G by a finite cut, contradicting that G is infinitely edge-connected.
Here is the second main lemma of this section: Lemma 5.8. Suppose that G is an infinitely edge-connected graph, that u, v are two distinct vertices of G, and that C is a component of G − u − v such that u sends at least one edge to C. IfC has a recursively prunable finitely separating rooted spanning tree T such that u sends no edges to parts t ∈ T that are finite-degree nodes of T , then G[C + u + v] contains an arrow barrage minor from u to v.
Proof. Given T , we let X ⊆ V (T ) consist of the 0-labelled nodes of T that are minimal in the tree-order. Then the nodes in X form a maximal antichain in the tree-order, giving T = X ∪ X , as T is recursively prunable. Note that all the nodes in X have degree at most two in T . We claim that X must be infinite. Indeed, if X is finite, then so is X , and in particular T is locally finite. But then u may send no edges to C by assumption, contradicting our other assumption that u does send an edge to C. Therefore, X must be infinite.
Recall that the finitely separating spanning tree T ⊆C gives rise to a B ℵ0 (C)-tree (T, α). For every x ∈ X let us write (A x , B x ) := α(xp x , x, p x ) for the predecessor p x of x in T . As u sends some edges to C, but none to the parts in X , there is a neighbour w of u in the part x∈X B x of the star σ : By Lemma 5.2 we find an infinite subset Y ⊆ X such that the part of the infinite substar σ := { (A y , B y ) | y ∈ Y } ⊆ σ is connected. Note that w is contained in the part of σ because the part of σ is included in the part of σ . We now find an arrow barrage minor from u to v in G[C + u + v] as follows. For the branch set of the nock we take the part of σ plus the vertex u. For the branch set of the head we take {v}. The payloads we let be modelled by the subgraphs C[y], one for every y ∈ Y (here, each C[y] is infinitely edge-connected and sends infinitely many edges in G to v by Lemma 5.7 and Y ⊆ X).

Football minors.
We are almost ready now to prove Theorem 5.13. But first, we prove an intermediate proposition, which requires the following lemma and definition: Lemma 5.9. If G is an infinitely edge-connected graph and G is obtained from G by contracting disjoint finite vertex sets that are possibly disconnected, then G is infinitely edge-connected as well.
Proof. To show that G is infinitely edge-connected, consider any distinct two vertices x and y of G , and choose verticesx ∈ x andy ∈ y of G. Now, in the infinitely edge-connected graph G we choose infinitely many pairwise edge-disjointx-y paths P 0 , P 1 , . . . as follows. To get started, choose P 0 arbitrarily. At step n > 0, consider all the branch sets that are met by some P k with k < n, and let X n be their union. Then X n is finite, and we let P n be anx-y path in G that avoids all the finitely many edges of G running inside X n . Now everyx-y path P n ⊆ G gives rise to some x-y path P n ⊆ G satisfying E(P n ) ⊆ E(P n ) by a slight abuse of notation. We claim that the paths P 0 , P 1 , . . . are all edge-disjoint. For this, consider any two paths P n and P m with n < m, and let e be any edge of P n . Then e, viewed as an edge of G, runs between two branch sets that P n meets because it uses e. Hence these two branch sets are both included in X m , and so P m does not use any of the edges running between them. In particular, P m does not use e. A football with endvertices u and v is an infinitely edge-connected graph G containing u and v such that G − u − v is again infinitely edge-connected.
When we say that some graph contains a football minor connecting two vertices x and y we mean that the graph contains a football minor with some endvertices u and v such that the branch set corresponding to u contains x and the branch set corresponding to v contains y (or vice versa).
A muscle with endvertices u and v is a graph G that is obtained from the vertices u and v by disjointly adding an infinitely edge-connected graph H and adding one u-H edge ux and one v-H edge vy such that x = y.
A muscle barrage with endvertices u and v is a countably infinite union n∈N G n of muscles G n with endvertices u and v such that G n and G m do not meet in any vertices other than u and v for all n = m.
Muscle (barrage) minors connecting two vertices are defined like for footballs.
Proposition 5.11. Suppose that G is an infinitely edge-connected graph, that u, v are two distinct vertices of G, and that C is a component of G − u − v to which both u and v do send some edges. Then at least one of the following assertions holds: contains a football minor connecting u and v; (iii) G[C + u + v] contains an arrow barrage minor either from u to v or from v to u; in particular, G[C + u + v] contains a muscle barrage minor connecting u and v.
Proof. We may assume that both u and v send infinitely many edges to C. Indeed, if-say-u sends only finitely many edges to C, then consider the infinitely edgeconnected graph G := G[C + v] and let u be one of the neighbours of u in C. If there is a component C of G − u − v to which both u and v send infinitely many edges, then we may replace G, u, v, C with G , u , v, C . Hence we may assume that there are infinitely many components C 0 , C 1 , . . . of G − u − v such that, without loss of generality, u sends only finitely many but at least one edge to each C n while v sends infinitely many edges to each C n . By Theorem 5.1, allC n have finitely separating spanning trees. If oneC n has a finitely separating spanning tree that contains a subdivision of the infinite binary tree, then Lemma 5.3 provides a T ℵ0 * t minor witnessing (i). Otherwise, by Proposition 5.6, everyC n has a rooted finitely separating spanning tree T n that is recursively prunable. Then we pick for every n a finite-degree node t n ∈ T n , and we let P n be a path in C n that links a neighbour of u to the subgraph C n [t n ] such that P n has only its endvertex x n in C n [t n ]. Now we obtain an arrow barrage minor in G[C + u + v] from u to v that is sought in (iii), as follows. For the branch set of the arrow barrage's nock we take {u, u } ∪ n∈N V (P nxn ). The arrows' payloads we let be modelled by the infinitely edge-connected subgraphs C [t n ] (see Lemma 5.7). And for the branch set of the arrow barrage's head we take {v} (that v sends infinitely many edges to each part t n is ensured by Lemma 5.7 and the assumption that u sends only finitely many edges to each C n ).
Therefore, we may assume that both u and v send infinitely many edges to C. By Theorem 5.1 we may let T be a finitely separating spanning tree ofC, rooted arbitrarily. We make the following two observations.
If T contains a subdivision of the infinite binary tree, then Lemma 5.3 yields a T ℵ0 * t minor giving (i).
If T has finite-degree nodes t u and t v (possibly t u = t v ) such that u sends infinitely many edges to the part t u ⊆ V (C) in G and v sends infinitely many edges to the part t v ⊆ V (C) in G, then we deduce (ii), as follows. By Lemma 5.7 both is a football subgraph connecting u and v. Otherwise t u and t v are distinct. Then we let P be any t u -t v path in C, and (G[t u + u] ∪ G[t v + v] ∪ P )/V (P ) is a football minor connecting u and v.
By these two observations and Proposition 5.6, we may assume that T is recursively prunable and that, without loss of generality, whenever t ∈ T has finite degree then v does send infinitely many edges to the part t ⊆ V (C) in G while u may send only finitely many edges to it.
If u sends edges in G to infinitely many parts t ∈ T that have finite degree in T , then we find an arrow barrage minor from u to v giving (iii), because v sends infinitely many edges to all of the infinitely edge-connected subgraphs C[t] (cf. Lemma 5.7) by our assumption above. Otherwise u sends, in total, only finitely many edges in G to the parts t ∈ T that have finite degree in T . Since u sends infinitely many edges in G to C, this means that we may assume without loss of generality that u sends no edges to the parts t ∈ T that have finite degree in T . Then Lemma 5.8 yields an arrow barrage minor from u to v giving (iii).
Now we have all we need to prove the main result of the section, Theorem 5.13. In its proof, we will face the construction of a minor in countably many steps. The following notation and lemma will help us to keep the technical side of this construction to the minimum.
Suppose that G and H are two graphs with H a minor of G. Then there are a vertex set U ⊆ V (G) and a surjection f : U → V (H) such that the preimages f −1 (x) ⊆ U form the branch sets of a model of H in G. A minor-map ϕ : G H formally is such a pair (U, f ). Given ϕ = (U, f ) we address U as V (ϕ) and we write ϕ = f by abuse of notation. Usually, we will abbreviate 'minor-map' as 'map'. If we are given two maps ϕ : G H and ϕ : H H , then these give rise to another map ψ : G H by letting V (ψ) := ϕ −1 (ϕ −1 (V (H )) and ψ := ϕ • (ϕ V (ψ)). On the notational side we write ϕ ϕ = ψ.
Lemma 5.12. If G 0 , G 1 , . . . and H 0 ⊆ H 1 ⊆ · · · are sequences of graphs H n ⊆ G n with maps ϕ n : G n G n+1 that restrict to the identity on H n , then G n∈N H n . Proof. Recursively, each map ϕ n : G n G n+1 gives rise to a mapφ n : G 0 G n+1 viaφ 0 := ϕ 0 andφ n+1 := ϕ n+1 φ n . For every n ∈ N we write V n x =φ −1 n (x) for all vertices x ∈ H n+1 . For every vertex x ∈ H := n∈N H n we denote by N (x) the least number n with x ∈ H n . As the maps ϕ n restrict to the identity on H n , for every vertex x ∈ H the vertex sets V n x form an ascending sequence V ⊆ · · · whose overall union we denote by V x . We claim that the vertex sets V x form the branch sets of an H minor in G.
Indeed, every branch set V x is non-empty and connected in G because all V n x are. If xy is an edge of H, then G contains a V n x -V n y edge as soon as xy ∈ H n , and this edge is a V x -V y edge due to the inclusions V n x ⊆ V x and V n y ⊆ V y . It remains to show that V x and V y are disjoint for distinct vertices x, y ∈ H. This follows at once from the vertex sets V n x and V n y being disjoint for all n and the definition of V x and V y as ascending unions of these vertex sets.
Finally, we prove the main result of the section: Theorem 5.13. Suppose that G is any infinitely edge-connected graph, that u, v are two distinct vertices of G, and that C is a component of G − u − v to which both u and v do send some edges. Then at least one of the following assertions holds: (i) G[C + u + v] contains a T ℵ0 * t minor; (ii) G[C + u + v] contains a football minor connecting u and v.
Proof. Assume for a contradiction that both (i) and (ii) fail. We will use Proposition 5.11 to find the following graph H as a minor in G := G[C + u + v]. Let T u be an ℵ 0 -regular tree with root r u , and let T v be a copy of T u that is disjoint from T u . We write r v for the root of T v . The graph H is obtained from the disjoint union of the two trees T u and T v by adding the perfect matching between their vertex sets that joins every vertex of T u to its copy in T v . For every number n ∈ N we write H n for the subgraph of H that is induced by the first n levels of T u together with the first n levels of T v . Thus, H = n∈N H n . Finding an H minor in G completes the proof, because H/T u is isomorphic to T ℵ0 * t.
A foresighted H n is a graph that is obtained from H n by adding for every edge xy ∈ H n that runs between the two nth levels of T u and T v a muscle barrage B xy having endvertices x and y such that B xy contains no vertices from H n other than x and y, and all muscle barrages added are pairwise disjoint.
By Lemma 5.12 it suffices to find a sequence G Ĥ 0 Ĥ 1 · · · of graphsĤ n that are foresighted H n with maps ϕ n :Ĥ n Ĥ n+1 that restrict to the identity on H n ⊆Ĥ n in order to find an H minor inĤ 0 G . To get started, we apply Proposition 5.11 to G, u, v, C to obtain in G a muscle barrage minor connecting u and v. By turning one of the muscles into an edge we obtainĤ 0 G .
At step n > 0, consider the muscle barrages B xy that turn H n intoĤ n . For every muscle M k xy of each of these muscle barrages B xy = k∈N M k xy we apply Proposition 5.11 in M := M k xy − x − y to the neighbours x and y of x and y in M k xy and some component of M − x − y to which both x and y send some edges to find a muscle barrage minor connecting x and y . By turning one muscle of each new barrage into an edge, we find ϕ n :Ĥ n Ĥ n+1 .

Proof of the main result
In this section we employ the main result of the previous section (Theorem 5.13) to prove the main result of this paper (Theorem 1).
Lemma 6.1. If A and B are two infinite vertex sets in a graph G that does not contain a subdivision of K ℵ0 , then there are vertices a ∈ A and b ∈ B plus a finite vertex set S ⊆ V (G) {a, b} such that S separates a and b in G − ab.
Proof. The absence of such an S for a pair a = b means that, inductively, we can find infinitely many independent a-b paths in G. So if there is no S for every pair a = b, then inductively we find a T K ℵ0,ℵ0 in G, and T K ℵ0 ⊆ T K ℵ0,ℵ0 (contradiction).
Lemma 6.2. Suppose that G is a football with endvertices u and v. If G does not contain a subdivision of K ℵ0 , then G contains an infinitely edge-connected graph H as a minor with branch sets V h (h ∈ H) such that u and v are contained in distinct branch sets V x and V y , respectively, and there is a finite vertex set S ⊆ V (H) {x, y} separating x and y in H.
Proof. Write C for the infinitely edge-connected graph G − u − v. We apply Lemma 6.1 in C to the infinite neighbourhoods N (u) and N (v) of u and v in G to obtain vertices a ∈ N (u) and b ∈ N (v) plus a finite vertex set S ⊆ V (C) {a, b} that separates a and b in C − ab. Then H can be obtained from the infinitely edge-connected graph G − ab as follows. We discard all the edges that are incident with u or v, except for the two edges ua and vb each of which we contract. Then H is infinitely edge-connected because it is isomorphic to the infinitely edge-connected graph C −ab. And the way we treated the edges at u and v ensures that S separates the two vertices {u, a} and {v, b} in H as desired.
Lemma 6.3. Suppose that G is an infinitely edge-connected graph and that u, v are two distinct vertices of G that are separated in G by some finite vertex set S ⊆ V (G) {u, v}. Then there exist induced subgraphs H u , H v ⊆ G containing u and v respectively, such that the following assertions hold: is finite, non-empty and connected in G; (ii) both H u /X and H v /X are infinitely edge-connected; (iii) X avoids u and v; (iv) uX is an edge of H u /X and vX is an edge of H v /X.
Proof. Given G, u, v, S let us write C u and C v for the distinct components of G − S that contain u and v respectively. For both w ∈ {u, v} we abbreviate ∼ G[Cw∪S] as ∼ w . As G is infinitely edge-connected, we infer that every ∼ w -class meets S. In particular, there are only finitely many ∼ w -classes in total, which means that each of the non-singleton classes induces an infinitely edge-connected subgraph of G. Let us write K u and K v for the infinitely edge-connected subgraphs induced by the classes containing u and v respectively, i.e., To find H u and H v , we distinguish two cases.
In the first case, K u and K v are disjoint. For both w ∈ {u, v}, the finite partition of V (C w ) ∪ S induced by ∼ w has only finitely many cross-edges. Since G is infinitely edge-connected, this means that we can find a (K u ∩ S)-(K v ∩ S) path P in G avoiding all these finitely many edges. Then P , as it may not use these edges, is a K u -K v path with endvertices in S. We let P w be a w-P path in K w for both w ∈ {u, v}. Letting H u := G[K u ∪ P ∪vP v ] and H v := G[K v ∪ P ∪ůP u ] completes this case with X = V (P u ∪ P ∪ P v ) {u, v} because the graph H w /X contains the spanning subgraph K w /V (ẘP w ), and K w /V (ẘP w ) is infinitely edge-connected by Lemma 5.9 and because K w is infinitely edge-connected.
In the second case, K u and K v meet in a vertex s ∈ S. We write D u for the component of K u − u containing s. In D u we pick a finite tree T that contains the finite intersection V (D u ) ∩ V (K v ) ⊆ S and contains a neighbour of u. Then T contains s but neither u nor v. We let P v be any v-s path in K v . Letting H u := G[D u ∪vP v + u] and H v := G[K v ∪ T ] completes this case with X = V (T ∪vP v ): On the one hand, the graph H u /X is infinitely edge-connected because it contains the spanning subgraph G[D u + u]/V (T ) which is infinitely edge-connected by Lemma 5.9 and the fact that G[D u + u] is an infinitely edge-connected subgraph of K u . On the other hand, the graph H v /X contains the spanning subgraph K v /Y for Y := (V (K v ) ∩ V (D u )) ∪ V (vP v ), and K v /Y is infinitely edgeconnected by Lemma 5.9 and because K v is infinitely edge-connected. Definition 6.4 (Plows). Suppose that u and v are two distinct vertices. A halfplow with endvertices u and v is an infinitely edge-connected graph containing the edge uv. A plow with endvertices u and v and head h is a union of two half-plows with end-vertices u, h and h, v that do not meet in any vertex other than h. Plow minors connecting some two vertices are defined like for footballs and muscles. Theorem 6.5. If G is an infinitely edge-connected graph and u, v are two distinct vertices of G, then at least one of the following two assertions holds: (i) G contains a T ℵ0 * t minor; (ii) G contains a plow minor connecting u and v.
Proof. Let G, u, v be given, we show ¬(i)→(ii). For this, suppose that G does not contain a T ℵ0 * t minor. By Theorem 5.13 and Lemma 6.2 we may assume that there is a finite vertex set S ⊆ V (G) {u, v} that separates u and v in G. Then applying Lemma 6.3 provides induced subgraphs H u , H v ⊆ G containing u and v respectively, such that the following assertions hold: is finite, non-empty and connected in G; • both H u /X and H v /X are infinitely edge-connected; • X avoids u and v; • uX is an edge of H u /X and vX is an edge of H v /X.
Then (H u ∪ H v )/X is a plow minor connecting u and v.
Theorem 1. Every infinitely edge-connected graph contains either the Farey graph or T ℵ0 * t as a minor.
Proof. If G contains T ℵ0 * t as a minor, then we are done. So let us suppose that G does not contain a T ℵ0 * t minor. Our task then is to find a Farey graph minor in G. By Lemma 2.1 it suffices to find a halved Farey graph minor. Call a graph a foresighted halved Farey graph of order n ∈ N if it is the union ofF n with infinitely edge-connected graphs A xy , one for every blue edge xy ∈F n , such that: (i) each A xy meetsF n precisely in x and y but xy / ∈ A xy ; (ii) every distinct two A e and A e meet precisely in the intersection e ∩ e of their corresponding edges (viewed as vertex sets).
By Lemma 5.12 it suffices to find a sequence H 0 , H 1 , . . . of foresighted halved Farey graphs of orders 0, 1, . . . with maps ϕ n : H n H n+1 that restrict to the identity on F n ⊆ H n to yield a halved Farey graph minor in G =: H 0 . To get started, pick any edge e of G, and note that G = H 0 is a foresighted halved Farey graph of order 0 when we rename e to the edge of whichF 0 = K 2 consists. At step n+1, suppose that we have already constructed H n ⊇F n , and consider the infinitely edge-connected graphs A xy that were added toF n to form H n . Theorem 6.5 yields in each A xy a plow minor with head h xy that connects x and y. These plowminors combine withF n and with each other to give a map ϕ n : H n H n+1 ⊇F n+1 that sends the branch set of every head h xy to the vertex v xy ∈F n+1 −F n that arises from the blue edge xy ∈F n in the recursive definition ofF n+1 .

Outlook
Here are two open problems that came to my mind.
Problem 7.1. Can Theorem 1 be strengthened to always find one of the two minors with finite branch sets?
Seymour and Thomas [20], together with Robertson [16,17], have characterised the graphs without K κ or T κ minors in terms of tree-decompositions and, alternatively, in terms of various other structures. Can their list be extended to include the Farey graph? Tree-decompositions might not be the right complementary structures for infinitely edge-connected substructures, but there might be other structures (e.g. B ℵ0 (G)-trees): Problem 7.2. Characterise the graphs without a Farey graph minor in terms of tree-decompositions or in terms of other structures.