Monochromatic cycle partitions in random graphs

Erd\H{o}s, Gy\'arf\'as and Pyber showed that every $r$-edge-coloured complete graph $K_n$ can be covered by $25 r^2 \log r$ vertex-disjoint monochromatic cycles (independent of $n$). Here, we extend their result to the setting of binomial random graphs. That is, we show that if $p = p(n) \geq \Omega(n^{-1/(2r)})$, then with high probability any $r$-edge-coloured $G(n,p)$ can be covered by at most $1000 r^4 \log r $ vertex-disjoint monochromatic cycles. This answers a question of Kor\'andi, Mousset, Nenadov, \v{S}kori\'{c} and Sudakov.


Introduction
An active line of current research concerns sparse random analogues of combinatorial theorems. An early example of this type of result was given by Rödl and Ruciński [23], who proved a random analogue of Ramsey's theorem. Similar results have been obtained for asymmetric and hypergraph Ramsey problems (see the survey of Conlon [6]). In this paper we investigate the random analogue for monochromatic cycle partitioning.
Given an edge-coloured graph, how many vertex-disjoint monochromatic cycles are necessary to cover its vertices? Such a collection of cycles is called a monochromatic cycle partition. 1 The problem of finding a partition with few cycles was first investigated for edge-coloured complete graphs. Erdős, Gyárfás and Pyber [8] proved that there is a function f (r) such that every r-edge-coloured complete graph K n admits a partition into at most f (r) monochromatic cycles (independent of n). In particular, they showed that f (r) ≤ 25r 2 log r and further conjectured that f (r) = r. The case r = 2 was also conjectured by Lehel in 1979. Luczak, Rödl and Szemerédi [21] showed that f (2) = 2 for large n. The bound on n was then reduced by Allen [1] (though still large). Finally, Bessy and Thomassé [5] showed that f (2) = 2 for all n by an elegant argument. For r = 3, Gyárfás, Ruszinkó, Sárközy and Szemerédi [11] showed that f (3) ≤ 17. For general r, the best known upper bound for f (r) is due to Gyárfás, Ruszinkó, Sárközy and Szemerédi [12], who showed that f (r) ≤ 100r log r (for large n). On the other hand, Pokrovskiy [22] disproved Erdős, Gyárfás and Pyber's conjecture by showing that f (r) > r for all r ≥ 3. The question of whether f (r) is linear in r is still open. There has also been a considerable interest in monochromatic cycle partitions of host graphs that are not complete. For instance, graphs with few missing edges [10]; small independence number [24]; large minimum degree [4,7,19] and bipartite graphs [13,18]. See a survey of Gyárfás [9] for further information.
In this paper, we consider monochromatic cycle partitions in binomial random graphs Γ ∼ G(n, p). The study of partitioning Γ ∼ G(n, p) into monochromatic subgraphs was first initiated by Bal and DeBiasio [3], who showed that if p ≥ C(log n/n) 1/3 , then with high probability (w.h.p.), any 2-edge-coloured Date: 19th July 2018. The research leading to these results was supported by EPSRC, grant no. EP/P002420/1. 1 For this paper, we allow a single vertex and an edge to be 'degenerate' cycles. 1 Γ ∼ G(n, p) admits a partition into two vertex-disjoint monochromatic trees. Recently, Kohayakawa, Mota and Schacht [15] showed that the same holds for p = ω((log n/n) 1/2 ). For more colours, Ebsen, Mota and Schnitzer (see [15,Proposition 4.1]) showed that w.h.p. there exists an r-edge-colouring of Γ ∼ G(n, p) with p ≥ (log n/n) 1/(r+1) , which cannot be partitioned by r vertex-disjoint monochromatic trees. Covering Γ ∼ G(n, p) by (not necessarily vertex-disjoint) monochromatic cycles was studied by Korándi, Mousset, Nenadov,Škorić and Sudakov [17], who showed that if p ≥ n 1/r+ε , then w.h.p. any r-edge-coloured Γ ∼ G(n, p) can be covered by O(r 8 log r) monochromatic cycles. The same authors asked whether one can prove a random analogue of Erdős, Gyárfás and Pyber's theorem, that is, any r-edge-colouring of Γ ∼ G(n, p) admits a partition into constantly many monochromatic cycles. In this paper, we give an affirmative answer with p = Ω(n −1/(2r) ). Theorem 1.1 (Main result). Let r ≥ 2 and p = p(n) ≥ 2 6 r 5 n −1/(2r) . Then w.h.p. the random graph Γ ∼ G(n, p) satisfies the following property. Any r-edge-colouring of Γ admits a partition into at most 1000r 4 log r monochromatic cycles.
It would be interesting to improve our bound on p. A construction of Bal and DeBiasio [3] shows that for p = o((r log n/n) 1/r ) w.h.p. there exists an r-edgecolouring of Γ ∼ G(n, p), which requires an unbounded number of monochromatic components (and in particular, cycles) to cover all vertices. In light of this, it seems natural to conjecture the threshold to be of order (log n/n) 1/r . We did not attempt to optimize the number of cycles needed in Theorem 1.1. Thus it is likely that our bound offers some room for improvement. Let f n,p (r) be the minimum number of cycles needed such that w.h.p. every r-edge-colouring of Γ ∼ G(n, p) admits a partition into at most f n,p (r) monochromatic cycles. Bal and DeBiasio [2] showed that f n,p (2) > 2 if p ≤ 1/2. Moreover, Korándi, Lang, Letzter and Pokrovskiy [16] recently constructed r-edge-coloured graphs on n vertices with minimum degree (1 − ε)n which cannot be partitioned into fewer than Ω(ε 2 r 2 ) monochromatic cycles. Thus together with our conjectured threshold, an immediate question would be if it is true that f n,p (r) = o(r 2 ) for p = O((log n/n) 1/r )?

Notation
Let G = (V, E) be a graph and U, W ⊆ V be disjoint subsets of vertices. We denote the complement of U in G by U = V \ U . We write G[U ] for the subgraph of G induced by U and G − U for G[U ]. We write e G (U, W ) for the number of edges in G with one vertex in U and one in W . For a vertex v ∈ V , we write N G (v) for its neighbourhood in G and deg We omit the subscript G when it is clear from the context. For a collection C of graphs, we write We write log a for the natural logarithm of a. For sakes of exposition, we will omit ceiling and floor signs, whenever it is not important for the argument.

Proof outline
We sketch the proof of Theorem 1.1. Consider an r-edge-coloured Γ ∼ G(n, p). It is not difficult to find a small family C almost of vertex-disjoint monochromatic cycles covering all but at most εn vertices. Indeed a standard approach, which was introduced by Luczak [20] and uses the (sparse) regularity lemma, allows us to reduce the problem of finding a large cycle in Γ to finding a large matching in the reduced graph R of a regular partition of Γ. Thus we can obtain C almost by finding a large matching in the union of 4r 2 monochromatic components of R.
The main difficulty in proving Theorem 1.1 is to cover these leftover vertices with vertex-disjoint cycles that are also disjoint from C almost . Since these leftover vertices can only be determined after edge-colouring Γ, we cannot fix their location prior to exposing the edges. In the following lemma, we show that every small vertex set can be covered by a small number vertex-disjoint monochromatic cycles. Moreover, almost all of its vertices will be in a predetermined vertex set U .
Note that there might be a small set W ′ ⊆ W , whose neighbourhood does not behave in the expected way (e.g. there are too few edges between W ′ and U ∪ W ). Hence, in order to cover W ′ , we will need to use some vertices outside U . We defer the proof of Lemmas 3.1 to Section 5. Our proof of Lemma 3.1 is based on arguments of Korándi, Mousset, Nenadov,Škorić and Sudakov [17]. However, some new ideas are needed to ensure vertex-disjoint cycles.
Recall that we require the cycles covering W to be disjoint from those covering the rest of the vertices, i.e. C almost . To deal with this, we ensure that C almost is 'robust'. Roughly speaking, even after deleting a few vertices of C almost , there is a monochromatic cycle partition C ′ almost on the remaining vertices. This strategy was introduced by Erdős, Gyárfás and Pyber [8] and has become fairly standard in the area. Our 'robustness' property is more general in the sense that we further allow a deletion of a small set outside of U . This is crucial to allow the deletion of a small vertex set outside of U , which is used to cover W ′ .
We will prove Lemma 3.2 in Section 6. (See the beginning of Section 6 for a sketch proof of Lemma 3.2.) We now prove Theorem 1.1 using these two lemmas.
Consider any r-edge-colouring of Γ. By Lemma 3.2, there exist disjoint vertex sets U and W with |U | ≥ βn and |W | ≤ ε 1 n. Moreover, for any sets U ′ ⊆ U and W ′ ⊆ V (Γ) with |W ′ | ≤ 2 18 r 9 /p r , the graph Γ − (W ∪ W ′ ∪ U ′ ) admits a partition into at most 4r 2 + 1 monochromatic cycles. By Lemma 3.1, there exists a collection C 1 of at most 900r 4 log r monochromatic disjoint cycles such that W ⊆ V (C 1 ) and , so |W ′ | ≤ 2 18 r 9 /p r . By the choice of W and U , the graph Γ − (W ∪ W ′ ∪ U ′ ) admits a partition C 2 into at most 4r 2 + 1 monochromatic cycles. Thus C 1 ∪ C 2 is a partition of Γ into at most 900r 4 log r + 4r 2 + 1 ≤ 1000r 4 log r monochromatic cycles.

Probabilistic tools
The exposition of our probabilistic tools follows the one of [17]. We will use the following Chernoff-type bounds on the tails of the binomial distribution.  . Fix 0 < α, β < 1 and let C = 6/(α 2 β) and D = 9/α 2 . Then, for every p = p(n) ∈ (0, 1), w.h.p. the random graph Γ ∼ G(n, p) satisfies the following property: For any two disjoint subsets X, Y ⊆ V (Γ), satisfying either of The following lemma plays a key role in our proof. It says that given any vertex set X, there exists a vertex small set Y such that every k-sets S ⊆ X ∪ Y has the expected number of common numbers in X.
Lemma 4.4. Let k ≥ 1, α, β ∈ (0, 1). Let K = 12k/(α 2 β) and p = p(n) ≥ (K log n/n) 1/k . Then w.h.p. the random graph Γ ∼ G(n, p) satisfies the following property. For any vertex set X with |X| ≥ βn, there exists a set Y ⊆ X of size at Proof. We first show that w.h.p. Γ ∼ G(n, p) satisfies ( * ) for any set X of at least βn vertices and any family M of at least To see this consider a set X with |X| ≥ βn and a family M of at least By taking the union bound of these events over all such sets X and families M , we see that the probability that ( * ) fails is at most Thus we can assume that Γ satisfies ( * ). We will deduce the lemma from this property. Suppose that we are given a vertex set X with |X| ≥ βn. Let H − be an auxiliary Let M − and M + be matchings of maximum size in H − and H + , respectively. Then ( * ) implies that |M − | + |M + | ≤ K/(kp k ). Let Y = S∈M − ∪M + S and note that |Y | ≤ K/p k . By maximality of M − and M + , every k-set S in X ∪ Y satisfies deg * (S, X) = (1 ± α)p k |X|, as desired.

Proof of Lemma 3.1
In this section we prove Lemma 3.1, that is, every small vertex set can be covered by few monochromatic cycles. We start by setting up a few auxiliary lemmas.
For every family L of ℓ disjoint pairs of vertices and every set Y of 3ℓ vertices that is disjoint from each pair in L, we have Then, for any r-edge-colouring of G, there is a collection C of at most 3r 2 monochromatic disjoint cycles, which together cover all but at most K/p vertices of W and V (C) ⊆ U ∪ W .
Note that Lemma 5.2 is a slight strengthening of Lemma 2.1 in [17], which was originally stated for random graphs Γ ∼ G(n, p). The proof of Lemma 2.1 in [17] only relies on the fact that w.h.p. Γ ∼ G(n, p) satisfies conditions (i)-(iii). Thus, we omit its proof.
Then, for any r-edge-colouring of G, there is a collection C of at most 400r 4 log r monochromatic disjoint cycles such that Proof. Consider any r-edge-colouring of G with colours [r]. Define an r-edgecoloured auxiliary (multi)graph H on W as follows. For each j ∈ [r], we add an edge of colour j between vertices v, v ′ ∈ W , if one of the following holds.
(a) There are at least 2t vertices u ∈ U such that vuv ′ is a path of colour j.
We claim that the independence number α(H) of H is bounded by 2r − 1. Let Thus, without loss of generality, we may assume that c i,j = j for i ∈ Proof of Lemma 3.1. Let K = 24r 9 /β. We claim that w.h.p. Γ ∼ G(n, p) satisfies the following. (d) For any vertex set X of size |X| ≥ βn, there is a set Y ⊆ X of size at most K/p r such that, for each k ∈ {1, r}, each k-set S ⊆ X ∪ Y satisfies deg * (S, X) = (1 ± 1/r 4 )p k |X|.
Here (a) is a straight forward consequence of Lemma 4.1; (b) follows from Lemma 4.2 with 1/4, β/(8r) playing the roles of α, β; (c) follows from Lemma 4.3; and (d) follows from two applications of Lemma 4.4 with k = 1, r and α = 1/r 4 . We will deduce the lemma from these properties. Consider any r-edge-colouring of Γ and any disjoint subsets of vertices U and W with |U | ≥ βn and |W | ≤ (β/400r) 4 n. By (d), there is a set W 1 ⊆ W with We first cover W 1 . Let t 1 = K/p r . Since |W 1 | ≤ K/p r = t 1 , it follows that, for any r-set S ⊆ W 1 , we have Note that (b) implies e(X, Y ) > 0 for all disjoint X, Y ⊆ V (Γ) with |X|, |Y | ≥ t 1 . By Lemma 5.3 with W 1 , W 1 , t 1 playing the roles of U, W, t, there exists a collection C 1 of at most 400r 4 log r monochromatic disjoint cycles, such that and, for each k ∈ {1, r} and every k-set S ⊆ W 2 , we have (Note that such partition can be obtained w.h.p. by assigning each u ∈ U ′ randomly independently to U 2 with probability 1/2 and to U 3 otherwise.) Now we cover the majority of the vertices of W 2 using U 2 . We apply Lemma 5.2 with U 2 , W 2 , β/4 playing the roles of U, W, β. (This is possible as the conditions (i), (ii) and (iii) of Lemma 5.2 are satisfied by properties (b), (c) and (5.2).) Thus there is a collection C 2 of at most 3r 2 monochromatic disjoint cycles, which together cover all but at most 16000r 4 /(βp) vertices of W 2 and V (C 2 ) ⊆ U 2 ∪ W 2 .
It remains to cover W 3 = W 2 \ V (C 2 ). Let t 2 = 16000r 4 /(βp). It follows that, for any r-set S ⊆ W 3 , Thus, as above, by Lemma 5.2 with U 3 , W 3 , t 2 playing the roles of U, W, t, there exists a collection C 3 of at most 400r 4 log r monochromatic disjoint cycles such that Hence there are at most 400r 4 log r + 3r 2 + 400r 4 log r ≤ 900r 4 log r monochromatic cycles C := C 1 ∪ C 2 ∪ C 3 , which together cover W . Note that . This finishes the proof of Lemma 3.1.

Proof of Lemma 3.2
In this section we prove Lemma 3.2, i.e. we find a collection of monochromatic cycles that robustly partitions most of the vertices. Before we set up definitions and tools, let us outline our approach. Consider an r-edge-coloured graph Γ ∼ G(n, p). We apply the sparse regularity lemma to obtain a balanced vertex partition V = {V i } i∈[t] of Γ in which almost every pair (V i , V j ) is (sparse) regular. This allows to define an r-edge-coloured reduced graph R on [t], that encodes which pairs are regular and dense in one (or more) of the colours. An easy calculation shows that we can take R to be the union of at most 4r 2 monochromatic components with δ(R) ≥ 2t/3. Thus R contains a perfect matching R ′ . We allocate a monochromatic cycle C i ⊆ Γ for each monochromatic component in R and such that C = {C i } i∈[4r 2 ] covers most vertices of each pair (V i , V j ) with ij ∈ E(R ′ ). Our plan is to embed these cycles disjointly using the sparse blow-up lemma of Allen, Böttcher, Hán, Kohayakawa and Person [1]. As usual, there is a small set W ⊆ V (Γ) whose vertices do not behave well enough to be included this way. We define the set U by selecting each vertex of V (Γ) \ W with probability 2 −11 . It remains to show that for any small set W ′ ⊆ V (Γ) and U ′ ⊆ U , the graph Γ − (W ∪ W ′ ∪ U ′ ) admits a partition into roughly 4r 2 monochromatic cycles. By choice of U and as W ′ is small, the remainder of V is still regular and fairly well balanced. However, the rest of V might not any more be compatible with our initial allocation. To deal with this, we reallocate some of the edges of C along regular pairs (V i , V j ) with ij / ∈ E(R ′ ). This way, the new allocation takes the changes in V into account and we can finish by applying the blow-up lemma.
6.1. Sparse regularity and the blow-up lemma. Our exposition of sparse regularity and the blow-up lemma closely follows [1]. Let G = (V, E) be a graph and A and B be disjoint subsets of V . For 0 < p < 1, we define the p-density of the pair (A, B) 2 Similarly, we define a pair to be a sparse super-regular pair if it is a sparse regular pair and satisfies a minimum degree condition.
if it is (ε, d, p)-regular and, for every u ∈ A and v ∈ B, we have We remark that the term (d − ε)p|B| is a natural lower bound in the above minimum degree condition by the following fact, which easily follows from the definition of regularity. The next lemma also follows from the definition of regular pairs.
We say that a graph G with density p is (η, D)-upper-uniform with density p, if, for all disjoint sets U and W with |U |, |W | ≥ η|V (G)|, we have e G (U, W ) ≤ Dp|U ||W |. We will use the following sparse regularity lemma. Lemma 6.4 (Sparse regularity lemma [1, Lemma 6.3]). For any real D, ε > 0, integers r and t 0 , there exist η = η 6.4 (D, ε, r, t 0 ) > 0 and t 1 = t 6.4 (D, ε, r, t 0 ) with the following property. Let Γ be an r-edge-coloured, (η, D)-upper-uniform graph with density p on at least t 0 vertices. Then there is a partition V = {V i } i∈[t] of V (Γ) with the following properties.
(c) all but at most εt 2 pairs (V i , V j ) are (ε, d, p)-regular in each of the r colours for some possibly different d.
The sparse blow-up lemma has the (reasonable) requirement that neighbourhoods of vertices inherit regularity. Definition 6.5 (Regularity inheritance). Let A, B and C be vertex sets in G ⊆ Γ, where A and B are disjoint and B and C are disjoint, but we do allow A = C. We say that (A, B, C) has one-sided (ε, d, p)-inheritance if for each u ∈ A, the pair N Γ (u, B), C is (ε, d, p)-regular.
The setting in which the blow-up lemma works is as follows. Let G and H be two graphs on the same number of vertices.
be partitions of V (G) and V (H), respectively. We call the parts V i of G clusters. We say that V and X are size-compatible if |V i | = |X i | for all i ∈ [t]. Moreover, for κ ≥ 1, we say that V is κ-balanced if there exists m ∈ N such that we have m ≤ |V i | ≤ κm for all i ∈ [t]. Our goal is to embed H into G respecting these partitions.
We will have two reduced graphs R and R ′ ⊆ R, where R ′ represents superregular pairs and R regular pairs. More precisely, we require the following properties of R and R ′ and the partitions V and X of G and H. Definition 6.7 (Reduced graphs and one-sided inheritance). Let R and R ′ be graphs on vertex set [t].
• (H, X ) is an R-partition if H[X] is empty for all X ∈ X and ij ∈ E(R) whenever e H (X i , X j ) > 0.
for all ij ∈ E(R). In this case we also say that R is a reduced graph of the partition V. • Next we define define the so-called "buffer sets" of vertices in H. Note that we restrict ourselves to the case when H is triangle-free. Definition 6.8 (Buffer sets). Let H be a triangle-free graph with vertex partition X = {X i } i∈ [t] . Let R ′ ⊆ R be graphs on vertex set [t]. Suppose that (H, X ) is an R-partition. We say the familyX = {X i } i∈[t] of subsetsX i ⊆ X i is an (α, R ′ )-buffer for H if and each x ∈X i , the first and second neighbourhoods of x go along R ′ , that is, for each xy, yz ∈ E(H) with y ∈ X j and z ∈ X k , we have ij, jk ∈ E(R ′ ).
Let R be a graph on t ≤ t 1 vertices and let R ′ ⊆ R be a spanning subgraph with , respectively, which have parts of size at least m ≥ n/(κt 1 ). LetX = {X i } i∈[t] be a family of subsets of V (H). Suppose that (i) H is triangle-free; (ii) ∆(H) ≤ ∆, (H, X ) is an R-partition,X is an (α, R ′ )-buffer for H; and (iii) (G, V) is an (ε, d, p)-regular R-partition, which is (ε, d, p)-super-regular on R ′ and has one-sided (ε, d, p)-inheritance on R ′ . Then there is an embedding ψ : 6.2. Preliminaries. The next lemma will enable us to find the few connected matching in the reduced graph covering all its vertices. Lemma 6.10. Let δ ≥ γ > 0 and G an r-edge-colouring of graph on t vertices with δ(R) ≥ δt. Then G contains a spanning subgraph H with δ(R) ≥ (δ − γ)t, which is the union of at most r 2 /γ monochromatic components.
Proof. Define R to be the union of all monochromatic components of G each of size at least γt/r vertices. Note that, for each colour, there are at most r/γ such components. Hence R consists of at most r 2 /γ monochromatic components. Since each vertex v is in at most r monochromatic components, we have deg R (v) ≥ δ(G) − r · γt/r. In particular, R spans G.
The following lemma will indicate how to embed the monochromatic cycles into Γ ∼ G(n, p) with respective to the monochromatic components in the reduced graph obtained above. Lemma 6.11. Let s, t, m ∈ N with m ≥ 90t 3 s and t even. Let R be a graph on [t] with δ(R) ≥ 2t/3 and R ′ ⊆ R be a perfect matching. Suppose that R is the union of edge-disjoint connected subgraphs R 1 , . . . , R s . Let {x i } i∈[t] be a family of integers satisfying m ≤ x i ≤ 10m/9. Then there is a graph H, a partition X = {X i } i∈[t] of V (H) and a familyX = {X i } i∈[t] of subsets of V (H) with the following properties.
(ii) H is triangle-free; (iii)X is an (1/50, R ′ )-buffer for H; and (iv) H is the union of vertex-disjoint cycles C 1 , . . . , C s and at most one isolated vertex such that (C k , X ) is an R k -partition for each k ∈ [s]. In particular, (H, X ) is an R-partition and ∆(H) ≤ 2.
Proof. Let X = {X i } i∈[t] be a family of disjoint vertex sets satisfying |X i | = x i . For each k ∈ [s], let G k be the graph on i∈V (G k ) X i which contains an edge vw precisely when v ∈ X i and w ∈ X j for some edge ij ∈ E(R k ). Thus G := k∈[s] G k is a union of edge-disjoint graphs and (G, X ) is an R-partition.
First, we claim that there exists disjoint cycles C ′ 1 , . . . , C ′ s in G such that, for each k ∈ [s], • C ′ k ⊆ G k and C ′ k is triangle-free; • for each edge ij ∈ E(R k ), there is a path P ij ⊆ C k which alternates between X i and X j and has order To see that such cycles exist, note that, by connectivity of R k , any two vertices of G k are connected by a path of order at most t. Thus at most t 3 vertices are needed for a cycle C ′ k in G k to "visit" all edges of R k , that is, the cycle C ′ k contains an edge between X i and X j for each ij ∈ E(R k ). Moreover, by replacing each edge of C ′ k with a path of length 3 (between the same clusters), we can guarantee triangle-freeness. Thus for each k ∈ [s] and ij ∈ E(R k ), C ′ k contains a path P ij of length 3 alternating between X i and X j and | k∈[s] V (C ′ k )| ≤ 2t 3 s. Since |X i | ≥ m ≥ 2t 3 s + 2m/45 and R ′ is a perfect matching, we further extends each P ij with ij ∈ E(R ′ ) to have length at least 4m/45. Thus our claims holds. Let By Dirac's theorem, G ′ contains a matching M , which misses at most 1 vertex. For each k ∈ [s], we obtain a cycle C k from C ′ k by extending each path P ij (with ij ∈ E(R k )) to include all edges of M between X i and X j .
Defining regular partition and reduced graph. Consider any r-edge-colouring of Γ with colours [r]. By (6.1), any disjoint sets X, Y ⊆ V (Γ) of cardinality at least ηn ≥ n/(2t 1 ) satisfy e Γ (X, Y ) ≤ 2p|X||Y |. This implies that Γ is (η, D)upper-uniform with density p. Therefore Lemma 6.4 guarantees a vertex partition (a 3 ) all but at most εt ′2 pairs (V i , V j ) are (ε, d, p)-regular in each of the r colours for some possibly different d. For a colour c ∈ [r], we denote by Γ c the spanning subgraph of Γ, which contains all edges of colour c. We define a r-edge-coloured reduced (multi)graph T on [t ′ ] with an edge ij of colour c if (V i , V j ) is an (ε, 1/r, p)-regular pair in Γ c . We reduce this to a simple r-edge-coloured graph on [t ′ ] by arbitrarily deleting all but one colours of each edge of T . Note that, by (6.1), by (a 3 ), the reduced graph T contains all but εt ′2 edges. It follows that there is a subgraph Without loss of generality, we can assume that V (T ′ ) = [t]. By Lemma 6.10, T ′ admits a spanning subgraph R with δ(R) ≥ 2t/3, which is the union of at most 4r 2 monochromatic components. Since t is even, R contains a perfect matching R ′ .
Define G ′ to be the spanning subgraph of Γ[ i∈[t] V i ] obtained by keeping an edge vw precisely when there is an edge ij in R such that v ∈ V i , w ∈ V j and vw has the same colour as ij.
Defining W . The set W will consist of four vertex sets i / ∈[t] V i , V exc , V deg and V inh . The removal of V exc ∪ V deg will ensure that (G, V) is super-regular on R ′ . The set V inh are vertices that do not inherit regularity.
This finishes the proof of Lemma 3.2.