On Ramsey-minimal infinite graphs

Let $F$, $G$ and $H$ be simple, possibly infinite graphs with no isolates. We say that $F$ is a $(G, H)$-arrowing graph if every red-blue coloring of its edges produces either a red $G$ or a blue $H$ in $F$. A $(G, H)$-arrowing graph $F$ is $(G, H)$-minimal if there is no proper subgraph $F' \subset F$ such that $F' \rightarrow (G, H)$. An important problem concerning finite graphs is determining whether the collection $\mathcal{R}(G, H)$ of all $(G, H)$-minimal graphs is finite (respectively, infinite). In this paper, we introduce a similar kind of problem for infinite graphs. We are interested in deciding whether a graph pair $(G, H)$ admits any $(G, H)$-minimal graphs. In other words, we would like to know whether $\mathcal{R}(G, H)$ is empty (respectively, nonempty). This problem has strong connections to the study of graphs which admit a nontrivial embedding, that is, infinite graphs which properly contain a copy of itself.


Introduction
Let F , G and H be possibly infinite, simple graphs with no isolated vertices. We follow some notation in [11]. We say that F arrows (G, H) or that F → (G, H) if for every red-blue coloring of the edges of F , there exists either a red G or a blue H contained in F . In this case, we say that F is an (G, H)-arrowing graph. A red-blue coloring of F is called (G, H)-good if F does not contain a red G or a blue H with respect to the coloring. An alternative definition for F → (G, H) would then be a graph F which admits no (G, H)-good coloring.
A (G, H)-arrowing graph F is said to be (G, H)-minimal if there is no proper subgraph F ′ ⊂ F such that F ′ → (G, H). In other words, F is (G, H)-minimal if it arrows (G, H) and F − e → (G, H) for every e ∈ E(F ). The collection of all (G, H)-minimal graphs is denoted as R(G, H). This collection satisfies the symmetric property R(G, H) = R(H, G).
The problem involving (G, H)-minimal graphs is classically done for finite G and H, as introduced in [5]. One of the major problems that arose was determining whether R(G, H) is finite or infinite. Following the studies done by Beardon [1] and Cáceres et al. [6] for magic labelings and the metric dimension of graphs, respectively, we attempt to extend this finite problem to an infinite one. To our knowledge, this is the first serious attempt to do so. It appears that some properties which are expected to be true for finite graphs do not hold in the scope of infinite graphs.
For finite graphs G and H, it is known that R(G, H) is nonempty. This is because we can obtain a (G, H)-minimal graph from an arbitrary (G, H)-arrowing graph by iteratively deleting enough edges. However, if one of G or H is infinite, 1 2 k Figure 1. A k-ray P k∞ .
then R(G, H) might be empty. As we shall see in Example 4.4, the ray P ∞ and K 2 as a pair do not admit any minimal graph. If we consider the double ray P 2∞ instead of P ∞ , we have that R(P 2∞ , K 2 ) = {P 2∞ }, and thus a minimal graph exists. An intriguing but difficult problem in general would be to classify which pairs (G, H) induce an empty (resp., nonempty) R(G, H). The study of Ramsey-minimal properties of infinite graphs is naturally related to graphs which are isomorphic to some proper subgraph of themselves. We will call such graphs self-embeddable. Note that if F is a self-embeddable graph, then we can pick an The notion of a self-embeddable graph differs from that of a self-contained graph [12], which is one isomorphic to a proper induced subgraph of itself. While selfcontained graphs have applications in other problems, such as the tree alternative conjecture [3], we do not require the proper subgraph to be induced in our case, hence the differing vocabulary.
The general outline of this paper is as follows. In Section 2, we present the notation and convention in common use for this paper. We then give some compactness results for Ramsey-minimal graphs in Section 3. In Section 4, we obtain some general progress on Problem 1.1. In Section 5, we turn our attention to the case where G is an infinite graph and H is a matching nK 2 . For an example of previous work on Ramsey-minimal finite graphs involving matchings, see Burr et al. [4].

Preliminaries
In this paper, we exclusively work with simple, countable (including finite) graphs G = (V (G), E(G)) with no isolated vertices (i.e. every vertex of G is adjacent to another vertex). For n ≥ 1, nG denotes the graph consisting of n disjoint copies of G.
We say that H is a subgraph of G (or simply On the other hand, we say that G (properly) contains H, or that H (properly) embeds into G, if there is a (proper) subgraph H ′ of G such that H ′ ∼ = H. The difference between H being the subgraph of G and H being embedded into G is a subtle one, but we highlight it here for clarity. v Figure 2. A self-embeddable graph G that is not strongly selfembeddable. The red subgraph illustrates the image of a nontrivial self-embedding of G in the form of a right translation by 1.
The ray P ∞ is an infinite graph of the form where x 0 is its endpoint. The double ray P 2∞ , on the other hand, is of the form ({x n : n ∈ Z}, {x n x n+1 : n ∈ Z}). In general, a k-ray P k∞ (shown in Figure 1), k ≥ 1, is formed by identifying k endpoints of k distinct rays.
A family of graphs of particular interest is the family of comb graphs. Let N = {1, 2, . . .} be the set of positive integers and let ℓ : N → N be a function. The comb C ℓ is a graph obtained from a base ray P ∞ (called the spine) by attaching, for every n, a path P ℓ(n) of order ℓ(n) by one of its endpoints to the vertex x n of P ∞ . Other types of infinite graphs of interest include the infinite complete graph K ∞ and the infinite star S ∞ = K 1,∞ .
We use some terminologies of embeddings from [2,9,10]. Recall that a graph homomorphism G → H is a map ϕ : If ϕ is injective, then the homomorphism is called an embedding G ֒→ H. An embedding G ֒→ G is said to be a self-embedding of G. A self-embedding is nontrivial if its image, seen as a graph with vertex set ϕ(V (G)) and edge set {ϕ(v)ϕ(w) : vw ∈ E(G)}, is a proper subgraph of G.
A graph G is said to be self-embeddable if it has a nontrivial self-embedding. In other words, a self-embeddable graph is a graph that properly embeds into itself. We say that G is strongly self-embeddable if it admits an embedding into G − v for every v ∈ V (G). A strongly self-embeddable graph is clearly self-embeddable, but the converse does not hold in general, as shown in the following example.
Example 2.1. The infinite star S ∞ is self-embeddable but not strongly so (since S ∞ does not embed into the null graph S ∞ − c, where c is the center vertex). Another example can be found in the graph G of Figure 2. It is self-embeddable, with a right translation as its nontrivial self-embedding. However, G does not embed into the disconnected graph G − v, where v is the vertex indicated in the figure.
By induction, we easily obtain the following stronger properties for strongly self-embeddable graphs.
Proposition 2.2. Let G be strongly self-embeddable. Then: Proof. (i) Suppose that V = {v 1 , . . . , v n } and that H = G−{v 1 , . . . , v n−1 } contains G. In other words, there is a subgraph G ′ ⊆ H isomorphic to G. Since G ′ and G are isomorphic, G ′ is strongly self-embeddable. Thus, G ′ −v n must contain G ′ ∼ = G.
(ii) Suppose that e = uv. Since G − v contains a copy of G by hypothesis, we have that G − e ⊇ G − v contains G as well. The statement for any G − E then follows by induction.

Compactness Results
The aim of this section is, for countable graphs F , G, H, to express what F → (G, H) means in terms of their finite subgraphsF ,Ĝ,Ĥ. We start by providing the following form of Kőnig's infinity lemma to pave way for the compactness arguments used in this section.
. . be an infinite sequence of disjoint nonempty finite sets, and let K be a graph on their union. Assume that every vertex in V n+1 has a neighbor in V n . Then, K contains a ray v 0 v 1 . . . such that v n ∈ V n for all n.
Theorem 3.2 confirms that all (G, H)-minimal graphs are finite if G and H are finite. This result assures us that we only need to deal with the case where one of G and H is infinite, since taking both as finite graphs would produce a completely finite problem.
Proof. The case where F is finite is trivial by choosing F ′ = F , so assume that it is countably infinite. Suppose no finiteF ⊆ F arrows (G, H). Let e 0 , e 1 , . . . be an enumeration of E(F ) and denoteF n as the subgraph of F induced by {e 0 , . . . , e n }. Define V n as the set of all (G, H)-good colorings ofF n . Each V n is nonempty by assumption, and has at most 2 n+1 elements.
Let K be the graph on ∞ n=0 V n , where we insert all edges between c ∈ V n+1 and its restriction c|F n ∈ V n . Since the restriction of a good coloring is also good, the hypotheses of Lemma 3.1 are satisfied. Therefore, there exists a ray of (G, H)-good colorings c 0 c 1 . . ., where c n ∈ V n for all n, such that c = ∞ n=0 c n is a well-defined coloring of F .
We claim that c is (G, H)-good. Let us assume the contrary. Without loss of generality, suppose that F has a red subgraph G ′ ∼ = G with respect to the coloring c. Since G ′ is finite, there exists a sufficiently large n such thatF n has G ′ as a subgraph. It follows that c n is not (G, H)-good; contradiction. Now, we want to be able to characterize embeddability of a graph G into another graph F in terms of embeddability of finite subgraphsĜ ⊆ G into F . Ideally, we would want to prove a result such as: G embeds into F if and only if every finite subgraphĜ ⊆ G embeds into F . Sadly, this result is not true; a counterexample is shown in Figure 3. The problem is that the embeddingsĜ ֒→ F are incompatible in some sense-larger finite subgraphsĜ ⊆ G must be embedded further down F , and so there is no way to "stitch together" these embeddings to get an embedding G ֒→ F . To ensure compatibility between partial embeddings, we instead work with the following notion of a pointed graph. We also need to ensure that our graphs are locally finite (that is, deg(v) < ∞ for each vertex v).
A pointed graph is a triple G = (V, E, * ), where (V, E) is a graph, and * ∈ V is a specified vertex of G, called the basepoint of G. A pointed subgraph H is a subgraph G F Figure 3. Every finite subgraphĜ ⊆ G embeds into F , but G itself does not.
of G such that * H = * G . A pointed homomorphism is a graph homomorphism mapping basepoints to basepoints-we call it a pointed embedding if it is injective. A pointed graph is locally finite, connected, etc. if the underlying graph is.
Using pointed graphs, we can obtain a version of our desired result: Let F and G be locally finite, pointed graphs and suppose G is connected. If every connected, finite, pointed subgraphĜ of G admits a pointed embedding into F , then G admits a pointed embedding into F .
Proof. Assume that G is countably infinite. Enumerate V (G) as follows: let v 0 = * , let v 1 , . . . , v k be the neighbors of * , let v k+1 , . . . , v ℓ be the vertices of distance 2 to * , etc. This indeed enumerates G by the assumption that G is locally finite and connected. Then, the induced subgraphsĜ n := G[v 0 , . . . , v n ] are all connected, finite, pointed subgraphs of G. For each n, let V n be the set of pointed embeddingsĜ n ֒→ F . Each V n is nonempty by assumption. Inductively, we show each V n is finite. We see that V 0 is finite, since there is a unique embeddingĜ 0 ֒→ F mapping * G to * F . Now assume V n is finite, and pick f ∈ V n . SinceĜ n+1 is connected, v n+1 is adjacent to some v j for j ≤ n. Since F is locally finite, f (v j ) has finite degree, so there are only finitely many ways to define f (v n+1 ) and extend f to an embedding in V n+1 . Since there are also only finitely many ways to choose f ∈ V n , it follows that V n+1 is finite, and so all the V n are finite by induction.
Similarly to before, let K be the graph on ∞ n=0 V n , where we insert all edges between f ∈ V n+1 and f |Ĝ n ∈ V n . By Lemma 3.1, K contains an infinite ray Interestingly, Lemma 3.3 actually generalizes Kőnig's lemma, in its more standard, graph-theoretical form (as found in [7, Proposition 8.2.1]). Proof. Pick an arbitrary basepoint * ∈ F . For every n, we claim that P n (with * Pn chosen as an endpoint) admits a pointed embedding into F . If P n does not admit a pointed embedding into F , then there is no vertex v ∈ V (F ) such that d( * F , v) ≥ n. Since F is connected and locally finite, we can enumerate V (F ) as follows: let v 0 = * , let v 1 , . . . , v k be the neighbors of * , let v k+1 , . . . , v ℓ be the vertices of distance 2 to * , etc. By stage n, we will have enumerated all of F , hence F is finite; contradiction. The result now follows from Lemma 3.3, with G = P ∞ and * P∞ as its endpoint.
For pointed graphs F , G, and a non-pointed graph H, we write  Proof. Take an arbitrary red-blue coloring of F such that F does not contain a blue H. Denote F ′ as the pointed subgraph of F induced by all the red edges. By assumption, all connected, finite, pointedĜ ⊆ G admits a pointed embedding into F ′ . By Lemma 3.3, G admits a pointed embedding into F ′ , so a red G exists as a pointed subgraph of F . Throughout this section and the next, we fix a pair of (potentially infinite) graphs G and H. We provide a sufficient condition under which R(G, H) is empty by first finding some suitable family of graphs F for the pair (G, H).

Theorem 4.1. Suppose that F is a (possibly infinite) collection of graphs such that:
(1) F → (G, H) for every F ∈ F ; (2) Every (G, H)-arrowing graph Γ contains some graph F ∈ F . We have the following: (i) R(G, H) ⊆ {F ∈ F : F is not self-embeddable}; (ii) If every F ∈ F is self-embeddable, then R(G, H) is empty.
Proof. (i) Fix a (G, H)-minimal graph Γ. By condition (2), there is an F ∈ F such that F is contained in Γ. Since F → (G, H) by condition (1), we must have F = Γ as Γ is (G, H)-minimal. Therefore, Γ ∈ F . By Observation 1.2, Γ is not self-embeddable, and we are done.
Conditions (1) and (2) are not sufficient to ensure that R(G, H) and {F ∈ F : F is not self-embeddable} coincide. For example, take G = P ∞ , H = K 2 and F = {P ∞ , P 2∞ }. While conditions (1) and (2) hold, R(P ∞ , K 2 ) can be shown to be empty while P 2∞ is not self-embeddable. Hence, R(G, H) ⊂ {F ∈ F : F is not self-embeddable}. That said, we can create an extra condition to make both sets equal.
Theorem 4.2. Let F be a collection of graphs such that conditions (1) and (2) of Theorem 4.1 hold. Suppose that we also have the following condition: (3) F 1 and F 2 do not contain each other for every different F 1 , F 2 ∈ F . We have the following: (1), it remains to show that no proper F ′ ⊂ F arrows (G, H). If we assume the contrary, then we have an F ′′ ∈ F contained in F ′ by condition (2). This implies that F contains F ′′ and contradicts condition (3).
The preceding theorems have a few applications. For example, in order to prove Theorem 5.4 of the next section, we will need to use Theorem 4.1(ii). Also, we can consider the special case where F is chosen as {G}. This yields the following results. (ii) Again, take F = {G}. Conditions (1) and (2) of Theorem 4.1 are both satisfied. Now we just need to show that G is self-embeddable. Color an arbitrary edge of G by blue and the rest of the edges by red. Since H = K 2 , there is no blue H in G. So by the fact that G → (G, H), there is a red copy of G in G. This proves that G properly contains itself, and thus self-embeddable.
The following examples demonstrate some direct applications of Theorem 4.3 for some pairs of graphs.

Some Results Involving Matchings
We saw in Theorem 4.3(i) an answer to Problem 1.1 whenever H = K 2 . Now, let us consider the more general case where H = nK 2 . It becomes apparent that the characteristics of R(G, nK 2 ), n ≥ 2, are still related to whether G is (strongly) self-embeddable.
Theorem 5.1. We have the following: (i) If G is connected and not self-embeddable, then for all n ≥ 2, we have nG ∈ R(G, nK 2 ) so that R(G, nK 2 ) is nonempty; (ii) If G is strongly self-embeddable, then R(G, nK 2 ) is empty for all n ≥ 2.
Proof. (i) Fix a red-blue coloring of nG which does not create a blue nK 2 . It follows that there must be a component of nG isomorphic to G which is colored all red. Hence, nG → (G, nK 2 ).
Now let e ∈ E(nG) be an arbitrary edge located in some component G ′ of nG. We show that nG − e → (G, nK 2 ) by constructing a (G, nK 2 )-good coloring of nG − e as follows: for every component of nG other than G ′ , color one of its edges by blue; color the rest of the edges by red. Since G is connected, there are exactly n − 1 components in nG other than G ′ , so this coloring only manages to produce a blue (n − 1)K 2 . Also, since G is not self-embeddable, there cannot be a red G in any of the components of nG. By the connectivity of G, there cannot be a red G in all of nG either. Therefore, this coloring is indeed (G, nK 2 )-good.
(ii) By appealing to Theorem 4.3(ii), it suffices to prove that G → (G, nK 2 ). Fix a red-blue coloring of G which does not create a blue nK 2 . We claim that this coloring creates a red G. We construct a set of vertices V ⊂ V (G) using the following algorithm: This algorithm must halt after at most n − 1 while loop iterations since the edge chosen at each iteration must be independent from the edges chosen at previous iterations. It is then clear that the output V is finite and that G − V only contains red edges. By Proposition 2.2(i), we have a copy of G in G − V . It follows that there exists a red copy of G in G, and we are done.
Remark 5.2. We note that Theorem 5.1(i) can fail to hold if G is disconnected. For example, given G = 2P 2∞ , it can be shown that (n + 1)P 2∞ → (G, nK 2 ). This implies that 2nP 2∞ is not minimal for n ≥ 2, so R(2P 2∞ , nK 2 ) cannot be shown to be nonempty using the previous line of reasoning.
Example 5.3. Observe that P ∞ is strongly self-embeddable, while P k∞ is connected and not self-embeddable for k > 1. By Theorem 5.1, we have for every n ≥ 2, R(P k∞ , nK 2 ) is empty if and only if k = 1.
We note that the converse of Theorem 5.1(ii) does not necessarily hold. There indeed exists a graph G = S ∞ not strongly self-embeddable such that R(G, nK 2 ) is empty for all n.
We prove Theorem 5.4 by first defining a collection of graphs F n such that conditions (1) and (2)  Proof. To prove condition (1), we show that F → (S ∞ , nK 2 ) for all F ∈ F n by induction on n. The base case n = 1 follows from the fact that F 1 = {S ∞ } and S ∞ → (S ∞ , nK 2 ). Now assume that every graph in F n arrows (S ∞ , nK 2 ), and let F ∈ F n+1 be arbitrary. Suppose that c is a red-blue coloring of F which produces no red S ∞ .
Pick an arbitrary vertex u adjacent to x n+1 ∈ X such that u / ∈ X and the edge x n+1 u is colored blue. This can be done since otherwise, x n+1 is incident to infinitely many red edges. Observe that F ′ := F − {x n+1 , u} is an element of F n , so it contains a blue nK 2 with respect to the coloring c| F ′ . Since this blue nK 2 and the blue x n+1 u form an independent set of edges, there exists a blue (n + 1)K 2 in F with respect to c. This proves that F → (S ∞ , (n + 1)K 2 ) and completes the induction. Now suppose that Γ arrows (S ∞ , nK 2 ). We claim that Γ contains at least n vertices of infinite degree. Assume that Y , where |Y | < n, is the set of vertices of Γ having an infinite degree. By coloring all edges incident to a vertex in Y by blue and the rest of the edges by red, we obtain a (S ∞ , nK 2 )-good coloring of Γ; contradiction. Now, we can take arbitrary vertices x 1 , . . . , x n of Γ of infinite degree. The induced subgraph F := Γ[x 1 , . . . , x n ] is an element of F n , therefore condition (2) holds.
Proof of Theorem 5.4. Let n ≥ 1. Take F n as defined in Lemma 5.5. Invoking Theorem 4.1(ii), we need to show that every F ∈ F n is self-embeddable.
Let F ∈ F n be arbitrary. We aim to define a nontrivial self-embedding ϕ of F . Denote U as the complement of X (i.e., U := V (F ) \ X), and for all 1 ≤ i ≤ n, let ρ i : U → {0, 1} be such that ρ i (u) = 1 iff u is adjacent to x i . Define ρ(u) := (ρ 1 (u), . . . , ρ n (u)) for u ∈ U . Since the image of ρ is finite, then by the infinite pigeonhole principle, there exists an x ∈ {0, 1} n such that ρ −1 (x) is an infinite set, It is clear that ϕ is injective, and it is nontrivial since u 1 / ∈ ϕ(V (F )). It remains to show that ϕ is a graph homomorphism. Suppose that uv ∈ E(F ). By condition (b), we can assume without loss of generality that v ∈ X, say v = x j for some 1 ≤ j ≤ n. Since v / ∈ U , we must have v / ∈ ρ −1 (x), so ϕ(v) = v. If u / ∈ ρ −1 (x), then ϕ(u)ϕ(v) = uv ∈ E(F ), and we are done. So assume that u = u i for some i ≥ 1. Since uv = u i x j is an edge, we have that ρ j (u i ) = 1. Thus, we also have ρ j (u i+1 ) = 1 since ρ(u i+1 ) = x = ρ(u i ). It follows that The proof that ϕ is a nontrivial self-embedding is then complete.
Now, let C ℓ be a comb with a ray x 0 x 1 . . . as its spine. Suppose that s C ℓ = min ℓ(n)>1 n exists (that is, C ℓ is not a ray). We can assume that our combs satisfy s C ℓ ≥ ℓ(s C ℓ ) − 1 without loss of generality. Observe that if s = s C ℓ is such that s < ℓ(s) − 1, then we can define a function so that C ℓ * ∼ = C ℓ and s * = s C ℓ * satisfies s * ≥ ℓ * (s * ) − 1. Here, C ℓ * is basically the comb obtained from C ℓ by exchanging the positions of x 0 . . . x s and P ℓ(s) in the comb.
The following theorem gives some equivalent statements to the statement that R(C ℓ , nK 2 ) is empty for all n ≥ 2. In particular, the theorem gives an answer for Problem 1.1 whenever G is a comb and H is a matching.
x 0 x 1 Figure 4. Two combs C ℓ such that (a) ℓ(n) = n, and (b) ℓ(1) = 3 and ℓ(n) = 2 for n > 1. For each of the two combs, the red subgraph illustrates the graph image of its nontrivial selfembedding.
We claim that j < k. Assume for the sake of contradiction that j = s = k. Since we have established that j ≤ ℓ(k) − 1 and s ≥ ℓ(s) − 1 = ℓ(k) − 1, we then have ℓ(k) − 1 = j = s = k. It follows that ϕ must be the map otherwise, which is not nontrivial; contradiction. Now we can define p = k − j ≥ 1. We prove that ℓ(n) ≤ ℓ(n + p) for all n ≥ s (the case where n < s is trivial since then ℓ(n) = 1).
In both cases, we see that statement 5 holds for the chosen p = k − j. This completes the proof.
Example 5.7. If ℓ(n) = n, then C ℓ satisfies s ≥ ℓ(s) − 1 (with s = 2) as well as statement 5 of Theorem 5.6 for any choice of p ≥ 1. As such, C ℓ is strongly selfembeddable via a positive translation. Figure 4(a) shows a positive translation of C ℓ by 1. In addition, we have that R(C ℓ , nK 2 ) is empty for all n ≥ 2 by Theorem 5.6.
The comb C ℓ * is illustrated in Figure 4(b) as a red subgraph of C ℓ . We see that R(C ℓ * , nK 2 ) is always empty since C ℓ * satisfies statement 5 of Theorem 5.6. By the fact that C ℓ ∼ = C ℓ * , we have that R(C ℓ * , nK 2 ) is always empty as well.

Concluding Remarks
Problem 1.1, in its full generality, is quite a challenging problem to attack. For this reason, we chose to devote a significant part of this study to the particular case where H is a matching. Even then, we were not able to completely answer Problem 1.1. While Theorem 5.1 managed to get us closer, we still have the following problem involving R(G, nK 2 ). Problem 6.1. Is there a sufficient and necessary condition for G under which R(G, nK 2 ) is empty for all n ≥ 2?
Further studies can also be done on other specific cases of the pair (G, H). Of course, another avenue of research would be to consider multi-color Ramseyminimal infinite graphs, as done in [8] for finite graphs.