Vertex covering with monochromatic pieces of few colours

In 1995, Erd\H{o}s and Gy\'arf\'as proved that in every $2$-colouring of the edges of $K_n$, there is a vertex cover by $2\sqrt{n}$ monochromatic paths of the same colour, which is optimal up to a constant factor. The main goal of this paper is to study the natural multi-colour generalization of this problem: given two positive integers $r,s$, what is the smallest number $\text{pc}_{r,s}(K_n)$ such that in every colouring of the edges of $K_n$ with $r$ colours, there exists a vertex cover of $K_n$ by $\text{pc}_{r,s}(K_n)$ monochromatic paths using altogether at most $s$ different colours? For fixed integers $r>s$ and as $n\to\infty$, we prove that $\text{pc}_{r,s}(K_n) = \Theta(n^{1/\chi})$, where $\chi=\max{\{1,2+2s-r\}}$ is the chromatic number of the Kneser gr aph $\text{KG}(r,r-s)$. More generally, if one replaces $K_n$ by an arbitrary $n$-vertex graph with fixed independence number $\alpha$, then we have $\text{pc}_{r,s}(G) = O(n^{1/\chi})$, where this time around $\chi$ is the chromatic number of the Kneser hypergraph $\text{KG}^{(\alpha+1)}(r,r-s)$. This result is tight in the sense that there exist graphs with independence number $\alpha$ for which $\text{pc}_{r,s}(G) = \Omega(n^{1/\chi})$. This is in sharp contrast to the case $r=s$, where it follows from a result of S\'ark\"ozy (2012) that $\text{pc}_{r,r}(G)$ depends only on $r$ and $\alpha$, but not on the number of vertices. We obtain similar results for the situation where instead of using paths, one wants to cover a graph with bounded independence number by monochromatic cycles, or a complete graph by monochromatic $d$-regular graphs.


Introduction
Call a subgraph of an edge-coloured graph monochromatic if all its edges have the same colour.This paper is concerned with the general problem of covering all the vertices of an edge-coloured graph by monochromatic pieces.To be more precise, suppose that F is a fixed family of graphs, containing the 'pieces' that we can use for the covering.A monochromatic F-covering of an edge-coloured graph G is then a collection of monochromatic subgraphs of G covering all the vertices, such that every subgraph in the collection is isomorphic to one of the graphs in F. Typical choices for F include the the collection F p of all paths or the collection F c of all cycles, where it is customary to consider single vertices and edges as degenerate cycles.Given a graph G, we are interested in finding monochromatic F-coverings that are as small as possible; for example, we might want to cover G using as few monochromatic paths or cycles as possible.
This type of problem goes back to a footnote in a 1967 paper of Gerencsér and Gyárfás [11] in which it is shown that in every colouring of the edges of the complete graph K n with two colours, one can find two monochromatic paths that form a partition of (and, in particular, a covering) all the vertices.Over the last fifty years, such problems have been studied in many variations, including for more than two colours [10,14,24], for various other choices of F (most notably for the family of cycles [2,6,13,16,23], but also for regular graphs [25], bounded-degree graphs [12], trees [1,7]), and for other choices of G (complete bipartite and multipartite graphs [7,14,17,26], graphs satisfying a minimum degree condition [5,8,21], random graphs [4,18,20], graphs with bounded independence number [5,27], . . .).We note that like the Gerencsér-Gyárfás result mentioned above, most (but not all) of these results apply to the stronger situation where one wants to partition the vertices of the graph into disjoint monochromatic pieces (as opposed to just covering the vertices).For more details we refer to the recent survey of Gyárfás [15].
The specific focus of this paper is on monochromatic F-coverings that altogether do not use too many different colours.For a collection S of monochromatic edge-coloured graphs, we denote by col(S) the total number of different colours used by the graphs in S.Then, given a graph G, a family F, and positive integers r and s, we will write c r,s (G, F) for the smallest number with the property that every r-colouring of the edges of G admits a monochromatic F-covering S such that |S| ≤ c r,s (G, F) and col(S) ≤ s.
For the simplest case where F = F p is the collection of paths, where there are only two colours, and where G is the complete graph, Erdős and Gyárfás [9] proved that It is open which of the two bounds (if any) is correct; Erdős and Gyárfás conjectured that the true value is √ n.In any case, we observe that this result is in stark contrast to the above-mentioned result of Gerencsér and Gyárfás [11], which implies that which is a constant independent of n.One goal of this project was to see how the result (1) generalizes to to other values of r and s.

Our results
In this paper, we restrict ourselves to graphs G with independence number at most α > 0.
We suppose that r, s, α are constants and that the size of G tends to infinity.Given r, s, α, we write c r,s,α (n, F) = max Thus c r,s,α (n, F) is the minimum integer k such that in every graph G with independence number at most α and every r-colouring of the edges of G, there exists a monochromatic F-covering S of G of size at most k that satisfies col(S) ≤ s.
To state our results, we must first recall the notion of a Kneser hypergraph.The Kneser hypergraph KG (α+1) (r, r − s) is the (α + 1)-uniform hypergraph on the vertex set [r]  r−s = {X ⊆ [r] : |X| = r − s} where the vertices X 1 , . . ., X α+1 ∈ [r]  r−s form a hyperedge if and only if they are pairwise disjoint as subsets of [r].A result of Alon, Frankl, and Lovász [3] states that the chromatic number of this hypergraph is Note that the range 1 ≤ s < αr/(α + 1) corresponds precisely to the case where KG (α+1) (r, r − s) has no edges.The case α = 1 (which corresponds here to the case where G = K n ) was conjectured by Kneser in 1955 and famously established by Lovász [22] in 1978 using topological methods.
Our first result gives a lower bound on c r,s,α (n, F).Note that there are certain trivial cases where c r,s,α (n, F) is very small simply because the graphs in F have many isolated vertices.To give an extreme example, if F contains for every n ≥ 0 the graph with n vertices and no edges, then trivially c r,s,α (n, F) = 1.The easiest way to avoid such issues is to insist that each graph in F has at most a bounded number of isolated vertices.In addition to this, we will assume that F is ∆-bounded, that is, that every graph in F has maximum degree at most ∆.Then we prove the following lower bound: Theorem 1 (Lower bound).Given any positive integers r, s, α, ∆, K such that r > s, there exists c > 0 such that the following holds.Let F be a ∆-bounded family of graphs with at most K isolated vertices each.Then for every n ∈ N, we have c r,s,α (n, F) ≥ cn 1/χ , where χ = χ(KG (α+1) (r, r − s)).
We remark that the conclusion of Theorem 1 fails when r = s; indeed, there are many situations where c r,r,α (n, F) is known to be constant.For example, Gyárfás, Ruszinkó, Sárközy, and Szemerédi [16] proved that c r,r,1 (n, F c ) ≤ 100r log r.Sárközy [27] proved that c r,r,α (n, F c ) ≤ 25(αr) 2 log(αr).Sárközy, Selkow, and Song [25] proved that if F contains the graph on a single vertex and all connected d-regular graphs, then c r,r,1 (n, F) ≤ 100r log r+2rd.For more general families, Grinshpun and Sárközy [12] showed that if F is ∆-bounded and contains at least one graph on i vertices for every We also prove an upper bound that matches the lower bound given by Theorem 1 in many cases.Note again that it is possible to choose F so that c r,s,α (n, F) is trivially very large; for example, if F only contains a single fixed graph then it is obvious that c r,s,α (n, F) = Ω(n).Our way to avoid this kind of problem will be to assume that there is some ε > 0 such that for every i ≥ 1, F contains at least one graph F with |V (F )| ∈ [εi, i].In fact, our proof (but perhaps not the result) requires the stronger assumption that at least one such graph is bipartite.We prove: Theorem 2 (Upper bound).Given any positive integers r, s, α, ∆ such that r > s, and any ε > 0, there exists C > 0 such that the following holds.Let F be a ∆-bounded family F of graphs such that for every i ≥ 1, there is a bipartite F ∈ F with εi ≤ |V (F )| ≤ i.Then for every n ∈ N, we have c r,s,α (n, F) ≤ Cn 1/χ + c r,r,α (n, F), where χ = χ(KG (α+1) (r, r − s)).
This upper bound coincides asymptotically with the lower bound given by Theorem 1 whenever we know that c r,r,α (n, F) = O(n 1/χ ).As mentioned above, in many situations it is even known that c r,r,α (n, F) = O(1).We can thus obtain asymptotically tight results in several different cases.From the above-mentioned result of Sárközy [27] we immediately obtain: Corollary 3 (Paths and cycles).Let r, s, α be fixed positive integers such that r > s.Let χ = χ(KG (α+1) (r, r − s)).Let F p be the family of all paths and F c be the family of all cycles. Then In particular, setting α = 1 and using (2) gives thus generalizing the Erdős-Gyárfás result (1) to more colours (and the same holds for F c instead of F p ).
Similarly, using the result of Sárközy, Selkow, and Song [25], we get the following result for covering complete graphs by regular graphs: Corollary 4 (d-regular graphs).Let r, s, d be fixed positive integers such that r > s.Let χ = χ(KG (2) (r, r − s)) = max {1, 2 + 2s − r}.Let F d be the family containing all connected d-regular graphs and also the graph with a single vertex and no edges.Then Note that the bounds in Corollaries 3 and 4 are only tight up to a large multiplicative factor depending on r, s, and α (resp.d).It would be interesting to determine these factors more precisely.As mentioned earlier, even the case where r = 2 and s = α = 1 is still open.
It is perhaps interesting to note that the proof of Theorem 2 does not actually use the Alon-Frankl-Lovász result (2), but rather works directly with the definition of χ as the chromatic number of KG (α+1) (r, r − s).On the other hand, our proof of Theorem 1 really uses the value of χ given by ( 2), or, more precisely, it uses the lower bound on χ implied by (2), which is by far the more difficult direction.

Notation
We write [k] = {1, . . ., k}.We write A ℓ for the set of all ℓ-element subsets of the set A. If G is a graph and V i , V j are disjoint subsets of the vertices of G, then we denote by G[V i , V j ] the bipartite subgraph induced by the two parts V i and V j , and we write e G (V i , V j ) for the number of edges of Since we are aiming for asymptotic statements, we routinely omit rounding brackets whenever they are not essential.
Suppose that we are given positive integers r, s, α, ∆, K such that r > s.Let χ denote the chromatic number of KG (α+1) (r, r − s).We need to show that there is a constant c = c(r, s, α, ∆, K) > 0 such that c r,s,α (n, F) ≥ cn 1/χ for all n ∈ N and all ∆-bounded families F of graphs with at most K isolated vertices each.
In other words, we need to construct an r-coloured graph G with independence number at most α such that every monochromatic F-covering S of G with col(S) ≤ s has size at least cn 1/χ .The construction will use Johnson graphs.The Johnson graph J(a, b) is the graph with the vertex set [a]  b where two vertices X and Y are joined by an edge if they have a non-empty intersection (so it is the complement of the Kneser graph KG (2) (a, b)).Is is easy to see that the independence number of J(a, b) is at most ⌊a/b⌋: every collection of ⌊a/b⌋ + 1 sets in [a]   b covers in total (⌊a/b⌋ + 1)b > a elements, counted with multiplicities, so that at least two of the sets must intersect.
To prove Theorem 1, we use different constructions depending on the parameters.We distinguish between three cases.
Case 1. Suppose first that 1 ≤ s < αr/(α+1), i.e., that χ = 1 by ( 2).Let G be a blow-up of J(r, r − s) where every vertex is replaced by a clique on n/ r r−s vertices and where every edge is replaced by a complete bipartite graph between the corresponding cliques.For a vertex X of J(r, r − s), we write V X for the vertices of G in the clique corresponding to X.
Note that G has the same independence number as J(r, r −s), which is at most ⌊r/(r −s)⌋.The assumption αr/(α + 1) > s implies that r r − s < r r − αr/(α + 1) = α + 1 and so the independence number of G is at most α, as required.We now colour the edges of G with colours from [r] as follows.Let uv be an edge of G. Then there exist vertices X and Y of J(r, r − s) such that u ∈ V X and v ∈ V Y .Moreover, we either have X = Y , or {X, Y } is an edge in J(r, r − s), and in both cases, X ∩ Y = ∅.We then colour uv with any colour belonging to the set Finally, suppose that S is a monochromatic F-covering of G such that col(S) ≤ s.Then there is some X ⊆ [r] of size r − s that is disjoint from the set of colors used by the graphs in S. By our choice of colouring, all edges touching V X have a colour in X, so the vertices in V X can only be covered using isolated vertices.Since every graph in S has at most K isolated vertices, this means that |S| ≥ |V X |/K ≥ n/(K r r−s ), completing the proof in this case (since χ = 1).
Case 2. Suppose now that s ≥ αr/(α + 1) and assume additionally that s < χα.Then by (2), we have χ = 1 + s − r + ⌈(s + 1)α −1 ⌉ ≤ r, where the last inequality follows from s + 1 ≤ r.Since additionally s + 1 ≤ χα, we can fix integers We now construct an n-vertex graph G as follows.We start with a blow-up of the complete graph K χ where the i-th vertex is replaced by a set V i of n i/χ vertices, except for the χ-th vertex, which is replaced by a set set V χ of vertices.Each edge ij of K χ is replaced by a complete bipartite graph between the corresponding sets V i and V j .We further partition each set V i equitably into k i parts V i,1 , . . ., V i,k i , and insert all edges where both endpoints are contained in the same set V i,j .Thus for each i, the graph G[V i ] is the disjoint union of k i cliques of size |V i |/k i .This defines the graph G.It is easy to see that G has independence number max {k i : Next, we colour the edges of G as follows.First, we fix an arbitrary bijection Such a bijection exists because Then we distinguish two cases.If uv is an edge of G with both endpoints in the same set V i,j , then uv receives the colour φ(i, j).On the other hand, if uv goes between the sets V i,j and V i ′ ,j ′ where i < i ′ , then we uv receives the colour φ(i, j).Note that by construction, there are no edges going between to sets V i,j and V i,j ′ for j = j ′ .Since k ≤ r, this is a colouring with at most r colours.Now suppose that S is a monochromatic F-covering of G such that col(S) ≤ s.Since s < k, there is then some pair (i, j) with 1 ≤ i ≤ χ and 1 ≤ j ≤ k i such that φ(i, j) is not the colour of any graph in S. Now observe that the only edges incident to V i,j that do not use the colour φ(i, j) are those that have an endpoint in In particular, every graph in S, having maximum degree at most ∆ and at most K isolated vertices, can cover at most and so to cover V i,j completely, S must contain at least (1 − o(1))n 1/χ /(r(∆ + K)) graphs, completing the proof in this case.Case 3. Finally, assume s ≥ αr/(α + 1) and s ≥ χα.The construction in this case is a combination of the constructions used in the two previous cases.We will construct a graph G on n vertices as follows.As in Case 2, we start with a blow-up of the complete graph K χ where the i-th vertex is replaced by a set V i of |V i | = n i/χ vertices, except for the last vertex, which is replaced by a set V χ of vertices.Each edge ij of K χ is replaced by a complete bipartite graph between the corresponding sets V i and V j .This defines the edges going between different sets V i and V j .
Next, we specify what each graph we use a similar construction as in Case 1.Let t := r − α(χ − 1) and note that since s ≥ χα > α(χ − 1), we have t > r − s.We let G[V 1 ] be a blow-up of the Johnson graph J(t, r − s) where every vertex is replaced by a clique on |V 1 |/ t r−s vertices, and where every edge is replaced by a complete bipartite graph between the corresponding cliques.For later reference, we define V 1,X ⊆ V 1 to be the vertex set of the clique corresponding to the vertex X of J(t, r − s).For 1 < i ≤ χ, we let G[V i ] be the union of α vertex-disjoint cliques of size |V i |/α, somewhat similarly as in Case 2. We will write V i,1 , . . ., V i,α ⊆ V i for the vertex sets of these cliques.This completes the definition of G.
We first check that G really has independence number at most α.It is immediate from the construction that α(G) = max {α(G[V i ]) : 1 ≤ i ≤ χ}.Moreover, it is easy to see that for i > 1, we have α(G[V i ]) = α.So it remains only to consider i = 1.Observe that G[V 1 ] has the same independence number as J(t, r − s), which is at most ⌊t/(r − s)⌋.It is thus sufficient to prove that t/(r − s) < α + 1, which is easily seen to be true using the definition of χ.Indeed, since t = r − α(χ − 1), the inequality t < (α + 1)(r − s) is equivalent to which is true because r − s + χ − 1 = ⌈(s + 1)/α⌉ using (2) and the assumption s ≥ αr/(α + 1).Hence we have α(G) ≤ α, as required.
We now define a colouring of the edges of G with r colours, where we distinguish several cases.First, suppose that uv is an edge with u, v ∈ V 1 .Then there exist vertices X, Y of J(t, r − s) such that u ∈ V 1,X and v ∈ V 1,Y ; moreover, for these X, Y it holds that X ∩ Y = ∅ (they are either identical or represent an edge in J(t, r − s)).We then colour uv with any colour in X ∩ Y .Second, assume that uv has exactly one endpoint (say, u) in V 1 and the other in V i for some i > 1.Then there is some vertex X of J(t, r − s) such that u ∈ V 1,X , and we colour uv with any colour in X. Lastly, to colour the remaining edges, fix any bijection Such a bijection exists because r − t = α(χ − 1).If uv is an edge with both endpoints in the same set V i for i > 1, say u, v ∈ V i,j , then we colour uv with the colour φ(i, j) (note that there are no edges between V i,j and V i,j ′ for j = j ′ ).If uv is an edge going between u ∈ V i,j and v ∈ V i ′ ,j ′ where i < i ′ , then we colour uv with the colour φ(i, j).Thus we have coloured all the edges.
We make two observations at this point: (i) Every edge incident to V 1,X is coloured with a colour from X, for every vertex X of J(t, r − s); (ii) For every 1 < i ≤ χ and 1 ≤ j ≤ α, the only edges incident to V i,j that do not use the colour φ(i, j) are those that are incident to a set V i ′ where i ′ < i.In particular, every monochromatic copy of a graph F ∈ F that uses a colour different from φ(i, j) can cover at most , where we use that F has maximum degree at most ∆ and at most K isolated vertices.
To complete the proof, suppose that S is a monochromatic F-covering of G such that col(S) ≤ s.Denoting by Col(S) the set of all colours used by graphs in S, we distinguish two possible cases.
The first case is when Col(S) contains at most t − (r − s) colours from [t].In this case, there is some set X of r − s colours in [t] that do not belong to Col(S).But then, as all edges incident to V 1,X use a colour from X (see (i)), the only way in which S can cover the vertices in V 1,X is by using isolated vertices.Since each graph in S has at most K isolated vertices, this implies |S| ≥ |V 1,X |/K ≥ n 1/χ /K, completing the proof in this case.
In the other case, Col(S) contains at least t − (r − s) In particular, there is a colour a ∈ [r] \ [t] that is not used by any of the graphs in S. Let (i, j) = φ −1 (a) and consider the set V i,j .Then by (ii), every graph in S can cover at most This completes the proof of Theorem 1.

Proof of Theorem 2
Let r, s, α be positive integers with r > s.Let K := KG (α+1) (r, r − s) and χ := χ(K).Let G be a graph on n vertices with independence number at most α, and suppose that the edges of G are coloured with r colours, which we assume to come from the set [r] = {1, . . ., r}.Then the vertices of K correspond naturally to sets of r − s colours.Let ∆, ε > 0 and let F be a ∆-bounded family of graphs with such that for every i ≥ 1, F contains at least one bipartite graph with at least εi and at most i vertices.In particular, F contains the graph on a single vertex and with no edges.We will show that there is a monochromatic where C = C(r, s, α, ε) > 0 is a suitable constant.We first note that if s < αr/(α + 1), then by (2), we have χ = 1.In this case, we can simply cover G by n single vertices, and we are done.Therefore, we will assume from now on that s ≥ αr/(α + 1).
We start by introducing some notation.If S is a monochromatic F-covering of G and X ∈ V (K) is a set of r − s colours, then we write V S,X ⊆ V (G) for the set of all vertices of G that are covered in S exclusively by graphs having a colour in X, that is, Note that S ⊆ S ′ implies V S ′ ,X ⊆ V S,X for all X ∈ V (K): adding more graphs to S can never increase one of the sets V S,X .Our goal will be to construct a small monochromatic F-covering S such that V S,X = ∅ for some X ∈ V (K).Note that in this case, G is completely covered by the graphs in S that have colours not in X, so by removing all graphs with a colour in X from S, we can obtain a monochromatic F-covering S ′ ⊆ S with col(S ′ ) ≤ s.
With this goal in mind, we define a quantity to track the sizes of the sets |V S,X |: where we can set δ(S) = −∞ if |V S,X | = 0 holds for some X ∈ V (K).Note that since |V S,X | ≤ n, we always have the bound δ(S) ≤ r r−s log n.Our central claim is: Claim 5.There is a constant β > 0 such that the following holds.If S is a monochromatic F-covering of G such that |V S,X | > n 1/χ for all X ∈ V (K), then G contains a (nonempty) collection H = {H 1 , . . ., H t } of monochromatic copies of graphs in F such that δ(S) − δ(S ∪ H) ≥ βtn −1/χ log n.
We postpone the proof of this claim and first show how it can serve to imply the theorem.We construct a monochromatic F-covering step by step, starting with some monochromatic F-covering S 0 of size c r,r (G, F) (which exists by definition).Then as long as |V S i ,X | > n 1/χ for all X ∈ V (K), we construct S i+1 from S i by setting S i+1 = S i ∪ H for a collection H as given by Claim 5. Note that since δ(S 0 ) ≤ r r−s log n, and since δ(S) ≤ 0 implies that |V S,X | ≤ 1 ≤ n 1/χ for some X ∈ V (K), it follows from (3) that this process must end after adding at most r r−s n 1/χ /β graphs to S 0 .In other words, we end up with a monochromatic F-covering S * of size |S * | ≤ c r,r (G, F)+ r r−s n 1/χ /β such that |V S * ,X | ≤ n 1/χ holds for at least one X ∈ V (K).From this we obtain another monochromatic F-covering S by adding to S * at most n 1/χ single-vertex graphs covering the vertices in V S * ,X .Note that then V S,X = ∅ and |S| ≤ c r,r (G, F)+ r r−s n 1/χ /β +n 1/χ .As mentioned above, we can then find a monochromatic F-covering S ′ ⊆ S with col(S ′ ) ≤ s, completing the proof of the theorem.

Proof of Claim 5
It remains to give the proof of Claim 5.The proof will use the following lemma, whose proof we omit (it is a standard application of Szemerédi's regularity lemma, see for example [19,Theorem 2.1]).Lemma 6.For every ε > 0 and ∆ > 0 there is a constant δ > 0 such that the following holds for all sufficiently large n.If G = (A, B, E) is a bipartite graph with |A| = |B| = n and |E| ≥ εn 2 , then it contains as a subgraph every bipartite graph with maximum degree at most ∆ and at most δn vertices.
In the following, let S be a monochromatic F-covering of G such that |V S,X | > n 1/χ for all X ∈ V (K).We first show: Claim 7.There exists a hyperedge E = {X 1 , . . ., X α+1 } of K such that Proof.Fix any c > 1 and let b ∈ (n 1/χ , cn 1/χ ) be such that b ≤ |V S,X | holds for all X ∈ V (K).This is possible because we assume that |V S,X | > n 1/χ for all X ∈ V (K).Then, because b ≤ |V S,X | ≤ n, the map X → ⌊log b |V S,X |⌋ assigns each vertex of K a number between 1 and ⌊log b n⌋ ≤ χ − 1.Hence, by definition of the chromatic number, there is a hyperedge E = {X 1 , . . ., X α+1 } in which all vertices receive the same number.Then for all i, j ∈ . Since c can be arbitrarily close to 1, and as K is finite, this implies the claim.
Let now E = {X 1 , . . ., X α+1 } be a hyperedge of K satisfying (4).We will assume the elements of E are ordered so that The idea behind this definition is that if W ⊆ V S,X i is removable, then by adding the graph H to S, we can decrease the size of |V S,X i | by at least |W |: indeed, recalling the definition of V S,X i , we see that V S∪{H},X i ⊆ V S,X i \ W . Claim 9.There is a constant C > 0 and some i ∈ [α + 1] such that the following holds: There Proof.Observe first that it is enough to show the following statement: for every choice of subsets V , . . ., V α+1 where V i ⊆ V S,X i and where each V i has size |V S,X α+1 |/2, there is some i ∈ [α + 1] and a subset W ⊆ V i of size at least |V S,X α+1 |/C that is removable.Indeed, we can then repeatedly apply this statement until we have covered all but |V S,X α+1 |/2 vertices in at least one set V S,X i , and it is clear that this requires at most C|V S,X i |/|V S,X α+1 | subsets of V S,X i .So we will now prove this other statement instead.
Fix sets V 1 , . . ., V α+1 as above.For brevity, write η : From the fact that G has independence number at most α it follows that there exist distinct i, j ∈ [α + 1] such that e G (V i , V j ) ≥ η 2 /(α + 1) 2 .This can be seen by simple double counting: for every choice of α + 1 vertices v i ∈ V i for i ∈ [α + 1], there must be two vertices that are connected by an edge.Going over all ways to choose such vertices, we thus obtain η α+1 edges, where every edge is obtained at most η α−1 times; so there must be η 2 edges going between the sets V 1 , . . ., V α+1 .In particular, for some i = j, we have e G (V i , V j ) ≥ η 2 /(α + 1) 2 .
Suppose now that e G (V i , V j ) ≥ η 2 /(α + 1) 2 .Let k ∈ [r] denote the majority colour of the edges in G[V i , V j ] and write G k [V i , V j ] for the subgraph consisting only of the edges having colour k.Then it is clear that G k [V i , V j ] has at least η 2 /(r(α + 1) 2 ) edges.
Recall that we assume that F is ∆-bounded and that there is some ε > 0 such that for every n ′ ≥ 1, the family F contains at least one bipartite subgraph F ∈ F with εn ′ ≤ |V (F )| ≤ n ′ .
Applying Lemma 6 to G k [V i , V j ] (which is possible for large n since |V i | = |V j | = η > n 1/χ /2) , we obtain that G k [V i , V j ] contains as a subgraph every ∆-bounded bipartite graph on at most 2(∆ + 1)η/(Cε) vertices, for some sufficiently large constant C > 0. In particular, G k [V i , V j ] contains a copy of a graph F ∈ F with at least 2(∆ + 1)η/C vertices.In fact, since F has maximum degree at most ∆, it can be embedded in such a way that is uses at least 2η/C vertices of V i and at least 2η/C vertices of V j (for every ∆ non-isolated vertices in V i we must embed at least one vertex in V j , whereas the isolated vertices can be embedded arbitrarily).Denote this copy by H and note that as a subgraph of G k [V i , V j ] it is clearly monochromatic in colour k.Since the sets X i and X j are disjoint (they are part of a hyperedge in K), they cannot both contain k, and so at least one of the sets V (H) ∩ V i or V (H) ∩ V j is removable, and both these sets have size 2η/C = |V S,X α+1 |/C.Let W 1 , . . ., W t ⊆ V S,X i be disjoint removable sets as in Claim 9 and let H = {H 1 , . . .H t } be the corresponding collection of subgraphs, so that H j is a monochromatic copy of a graph in F that covers W j and uses a colour outside X i .By Claim 9 and the definition of removable, we have |V S∪H,X i | ≤ |V S,X α+1 |/2 < |V S,X i |.This implies immediately that the collection H is nonempty.It also implies that δ(S ∪ H) = At the same time, using j∈[α+1] log |V S∪H,X j | ≤ j∈[α+1]\{i} log |V S,X j | + log |V S∪H,X i | ≤ δ(S) − log |V S,X i | + log(|V S,X α+1 |/2) = δ(S) − log(2|V S,X i |/|V S,X α+1 |),and so δ(S) − δ(S ∪ H) ≥ log(2|V S,X i |/|V S,X α+1 |).