Partitioning infinite hypergraphs into few monochromatic Berge-paths

Extending a result of Rado to hypergraphs, we prove that for all $s, k, t \in \mathbb{N}$ with $k \geq t \geq 2$, the vertices of every $r = s(k-t+1)$-edge-coloured countably infinite complete $k$-graph can be partitioned into the cores of at most $s$ monochromatic $t$-tight Berge-paths of different colours. We further describe a construction showing that this result is best possible.


Introduction
Lehel's conjecture (first seen in [2]) states that the vertices of every 2-edge-coloured complete graph can be partitioned into two monochromatic cycles, one of each colour. Here, single vertices and edges are considered to be cycles and this convention is used throughout this paper. The conjecture was proved for very large graphs by Łuczak, Rödl and Szemerédi [14] in 1998, for large graphs by Allen [1] in 2008, and for all graphs by Bessy and Thomassé [3] in 2010.
Erdős, Gyárfás and Pyber [6] conjectured in 1991 that this can be extended to r colours (allowing r monochromatic cycles). This was however disproved by Pokrovskiy [17], who showed that for every r ≥ 3, there are infinitely many r -edge-coloured complete graphs whose vertices cannot be covered by r monochromatic vertex-disjoint cycles. Finding the minimum number of monochromatic vertex-disjoint cycles needed to cover the vertices of any r -edge-coloured complete graph remains a big open problem. We note that a priori it is not obvious that this minimum is independent of the size of the complete graph we wish to cover. The fact that this is the case was proved by Erdős, Gyárfás and Pyber [6], who also presented a simple construction in which r cycles are needed. The currently best-known upper bound is 100r log r , due to Gyárfás, Ruszinkó, Sárközy and Szemerédi [9].
An infinite analogue of the conjecture of Erdős, Gyárfás and Pyber is true as proved by Rado [18] already in 1978.
Theorem 1.1 ([18]). The vertices of every countably infinite r -edge-coloured complete graph can be partitioned into r monochromatic paths (infinite or finite), one of each colour.
Rado's theorem is best possible, as the construction for finite graphs in [6] can be extended to infinite graphs.
In this note we consider extensions of this result to hypergraphs. A k-uniform hypergraph (or shortly, a k-graph) is a tuple There are different notions of paths in hypergraphs, loose paths, tight paths and Bergepaths; all of these coincide with the notion of paths when k = 2. In this note, we will mainly consider Berge-paths. Given integers 2 ≤ t ≤ k and ℓ ≥ 1, a finite (k-uniform) t-tight Berge-path of length ℓ is a pair (X , F ) defined as follows.
A 2-tight Berge-path is simply called a Berge-path and a k-tight Berge-path is called a tight path. The k-uniform loose path of length ℓ consists of n = k(ℓ − 1) + 1 vertices Many extensions of path partition problems to hypergraphs have been studied, considering loose paths [10,11,20], tight paths [5,4] and Berge-paths [10]. Most relevant for the topic of this note are the following two extensions of Theorem 1.1. Theorem 1.2 (Gyárfás-Sárközy [10]). The vertices of every countably infinite r -edgecoloured complete k-graph can be partitioned into r monochromatic loose paths (infinite or finite), one of each colour. Theorem 1.3 (Elekes-Soukup-Soukup-Szentmiklóssy [5]). The vertices of every countably infinite r -edge-coloured complete k-graph can be partitioned into r monochromatic tight paths (infinite or finite), one of each colour.
The latter result answers a question of Gyárfás and Sárközy from [10]. Note that both Theorems 1.2 and 1.3 reduce to Theorem 1.1 when k = 2. Our main result extends Theorem 1.1 in a similar way to Berge-paths. It turns out that ⌈r /(k − 1)⌉ paths suffice. Theorem 1.4. For all s, k ∈ N with k ≥ 2 and every r = s(k − 1)-edge-colouring of K (k) N , the vertices can be core-partitioned into s monochromatic Berge-paths of different colours.
Note that Theorem 1.4 reduces to Theorem 1.1 when k = 2 as well. We shall actually prove the following more general result about t-tight Berge-paths.
N , the vertices can be core-partitioned into s monochromatic t-tight Berge-paths of different colours.
Note that the case k = 2 reduces to Theorem 1.4, and the case k = t reduces to Theorem 1.3. The following theorem shows that Theorem 1.5 is best possible.
colours in which the vertices cannot be covered by the cores of s monochromatic t-tight Berge-paths.
We will prove Theorem 1.6 in Section 2 and Theorem 1.5 in Section 3.
2 Proof of Theorem 1.6 The construction described in the proof generalises the construction from [6].
Proof of Theorem 1.6. We denote the lexicographical ordering of For every e ∈ E(K (k) N ) we consider an order x 1 e , . . . , x k e of e satisfying I (x i e ) I (x j e ) for all 1 ≤ i < j ≤ k, and define φ(e) as an arbitrary member of [r ] \ i≤k−t+1 I (x i e ). Assume for contradiction that there are monochromatic t-tight Berge-paths P 1 , . . . , P s with cores X 1 , . . . , X s so that i X i = N and let C ⊆ [r ] be a set of size s which contains all colours used by these t-tight Berge-paths.
Observe that for every edge e with e ∩ B C ∅ and φ(e) ∈ C we have ) and thus φ(e) C. For i ∈ C, let F i be the set of all f ∈ X i t which consist of t consecutive vertices of X i with at least one element in B C . Let f ∈ F i and let e ∈ E(P i ) be some edge with f ⊆ e. By (2.1), we have e \ ≺C B ≤ t − 1 and therefore some vertex in f must be in ≺C B . Since every ∈ N is contained in at most t sets f ∈ F i , it follows that Observe now that for all but at most t − 1 vertices ∈ X i ∩ B C , there is a unique f ∈ F i starting at and thus Combining (2.2) and (2.3), we get 3 Proof of Theorem 1.5 Our proof is based on ideas from [5]. First, we need to introduce some notation. An rmulti-colouring of a k-graph G is a function χ : E(G) → 2 [r ] . Given a set F ⊆ E(G), we denote by χ (F ) = e∈F χ (e) the set of colours they have in common and say that F is (χ -)monochromatic if χ (F ) is non-empty. For a given r -colouring φ of K := K (k) N and i, j ∈ N with j < k, we define an r -multi-colouring φ i,j : N\{i} Furthermore, we call {K i : i ∈ N} a j-clique-chain w.r.t. an r -colouring φ of K if K 1 is a φ 1,j -monochromatic copy of K (j) Observe that, by Ramsey's theorem [19] for infinite hypergraphs, there exists a j-cliquechain for every r -colouring of K and every j ∈ [k − 1].
For a j-clique-chain {K i : i ∈ N} we define a vertex-multi-colouring χ : N → 2 [r ] by χ (i) = e∈E(K i ) φ i,j (e) for every i ∈ N. We call χ the clique-colouring induced by {K i : i ∈ N}.
Lemma 3.1. For all s, k, t ∈ N with k ≥ t ≥ 2 and every r = s(k − t + 1)-colouring of K (k) N , there is a (t − 1)-clique-chain that induces a clique-colouring using at most s colours.
Proof. Let φ be the given r -colouring of K := K (k) N . Furthermore, let C 1 ∪ · · · ∪ C r be a partition of the set of r = s(k − t + 1) colours into r blocks of size s. We will show that there is a (t − 1)-clique-chain and some i ∈ [r ] such that for the induced clique-colouring we have χ ( ) ∩ C i ∅ for all ∈ N.
We call an infinite (t − 1)-uniform clique K ′ maximally-monochromatic w.r.t. a multicolouring ψ of K ′ and a set C ⊆ [r ] if there is no infinite clique K ′′ ⊆ K ′ with | {i : ψ (K ′′ ) ∩ C i ∅} ∩ C| > | {i : ψ (K ′ ) ∩ C i ∅} ∩ C|. Note that a maximallymonochromatic clique is not necessarily monochromatic (since all its infinite monochromatic subcliques might have colours not in C). Further note that every infinite clique contains a maximally-monochromatic infinite clique (since r is finite).
We proceed now with the proof of Theorem 1.5.
Proof of Theorem 1.5. Let φ be the given r -colouring of K = K (k) N . By Lemma 3.1, there is a (t − 1)-clique-chain {K i : i ∈ N} that induces a clique-colouring χ using at most s colours (without loss of generality these colours are 1, . . . , s). For i ∈ [s], let A i ⊆ N be the set of vertices of colour i according to χ .
By repeating the following process we will simultaneously build t-tight monochromatic Berge-paths P 1 , . . . , P r with core-vertex sequences {b i,1 , b i,2 , . . .} for every i ∈ [s]. Let b i,1 := min A i for every i ∈ [s]. In every step, we will add to each path t or t − 1 vertices making sure that for every i ∈ [s], the last new vertex, say b i,n i , is in A i , and that the other new vertices are in V (K b i,n i ). Right after choosing the vertex b i,j , we will choose a unique edge e i,j ∈ E(K) of colour i which contains the t consecutive vertices b i,j−t+1 , . . . , b i,j for every j ≥ t and i ∈ [s]. Let X = {b 1,1 , . . . , b s,1 } and let Y = ∅. We will use X to keep track of already used vertices and Y to keep track of already used edges.
For each i ∈ [s] do the following. 2 Suppose the current path P i ends in b i,n ∈ A i for some n ∈ N. We will now extend P i by t or t − 1 vertices as follows. Let a be the smallest vertex in A i \ X (if A i \ X is empty, the path P i is complete and we move to the next step). Add a to X and do the following for every j = 1, . . . , t − 2. Choose a vertex b i,n+j ∈ V (K a ) \ ( Y ) and add it to X (note that this is always possible since V (K a ) is infinite and Y is finite). Let f i,n+j = {b i,n+j−t+1 , . . . , b i,n+j } be the set of the t consecutive vertices in the core of the Berge-path P i ending at b i,n+j . Note that f i,n+j \ {b i,n } ∈ E(K b i,n ) and thus i ∈ φ b i,n ,t−1 (f i,n+j \ {b i,n }). Hence, by the definition of φ b i,n ,t−1 , there exist e i,n+j ∈ E(K) with f i,n+j ⊆ e i,n+j and φ(e i,n+j ) = i. Add e i,n+j to Y . Since b i,n+j Y , we have e i,n+j Y and we can therefore use e i,n+j as the edge for f i,n+j in our Berge-path.
Choosing the next vertex will be slightly more complicated (since a might be in some edge in Y ). By construction, P 1 , . . . , P s are monochromatic t-tight Berge-paths whose cores are disjoint. Furthermore, since at the beginning of every step the smallest uncovered vertex a of A i is chosen, we have i V (P i ) = N.

Further remarks and open problems
Theorems 1.1 to 1.4 remain true when we consider cycles instead of paths, where an infinite cycle is a two-way infinite path. It is not clear to us however if one can replace paths by cycles in Theorem 1.5 when 2 < t < k. Difficulties only arise when trying to close finite paths to cycles, hence we can replace paths by cycles if we allow finitely many vertices to be uncovered.
A natural question to ask is if similar results hold in the finite setting. Gyárfás, Lehel, Sárközy and Schelp [8] conjectured that every finite (k − 1)-edge-coloured complete k-graph contains a monochromatic Hamiltonian Berge-cycle. Note that, in the infinite setting, this is a special case of Theorem 1.4. After partial results in [8,12,13,15], Omidi [16] announced a proof of this conjecture.
We believe that a generalisation of this to more colours, similar as in Theorem 1.4, is true as well.
Conjecture 4.1. For all s, k ∈ N with k ≥ 2, there is some c = c(s, k) ∈ N such that the following is true for all n ∈ N. In every r = s(k − 1)-edge-colouring of K (k) n , there is a collection of at most s monochromatic t-tight Berge-cycles whose cores are disjoint and cover all but c vertices.
For k = 2, this reduces to a conjecture of Pokrovskiy [17]. We further believe that this can be extended to t-tight Berge-cycles similarly to Theorem 1.5.

Conjecture 4.2.
For all s, k, t ∈ N with k ≥ t ≥ 2, there is some c = c(s, k, t) ∈ N such that the following is true for all n ∈ N. In every r = s(k − t + 1)-edge-colouring of K (k) n , there is a collection of at most s monochromatic t-tight Berge-cycles whose cores are disjoint and cover all but c vertices.
A simple modification of the construction in Section 2 shows that these conjectures are best possible (if true) apart from the finite leftover.