The extremal number of longer subdivisions

TL;DR

This paper proves conjectures regarding the extremal number of longer subdivisions of multigraphs, establishing bounds that depend on the subdivision length and confirming the predicted growth rates.

Contribution

It confirms Conlon and Lee's conjectures by proving the extremal number bounds for longer subdivisions of multigraphs and simple graphs.

Findings

Proved the extremal number bounds for the $(k-1)$-subdivision of multigraphs.

Confirmed the conjecture for simple graphs with a positive epsilon gap.

Established that the extremal number grows as $O(n^{1+1/k})$ for even $k$.

Abstract

For a multigraph $F$, the $k$-subdivision of $F$ is the graph obtained by replacing the edges of $F$ with pairwise internally vertex-disjoint paths of length $k+1$. Conlon and Lee conjectured that if $k$ is even, then the $(k-1)$-subdivision of any multigraph has extremal number $O(n^{1+\frac{1}{k}})$, and moreover, that for any simple graph $F$ there exists $\varepsilon>0$ such that the $(k-1)$-subdivision of $F$ has extremal number $O(n^{1+\frac{1}{k}-\varepsilon})$. In this paper, we prove both conjectures.

Equations

$$\mathrm{ex}(n,K_t^{k-1})=O(n^{1+\frac{16}{k}}).$$

$$\mathrm{ex}(n,F^{k-1})=O(n^{1+\frac{1}{k}}).$$

$$\mathrm{ex}(n,F^{k-1})=O(n^{1+\frac{1}{k}-\epsilon }).$$

$$\mathrm{ex}(n,F^{k-1})=O(n^{1+\frac{2}{k}-\epsilon }).$$

$$\mathrm{ex}(n,F^{k-1})=O(n^{1+\frac{1}{k}}).$$

$$\mathrm{ex}(n,F^{k-1})=O(n^{1+\frac{1}{k}-\epsilon }).$$

$$3\cdot 100^k|V(H)|\cdot c_{\ell }\le \frac{1}{2k}-\epsilon$$

$$n^{c_{\ell }}\le n^{\frac{1}{4k}-\epsilon /2}=o({\delta }^{1/2}).$$

$$X^{\prime }=\{x\in X:(x,y_{j_1})\text{ and }(x,y_{j_2})\text{ are }(\ell ,L)\text{-bad}\}$$

$$|X|\ge \frac{\delta }{f(\ell -1,L{)}^2}\ge \frac{L^{100^k|V(H)|}}{f(\ell -1,L{)}^2}\ge \frac{f(\ell -1,L{)}^{20|V(H)|}}{f(\ell -1,L{)}^2}\ge 2L(16f(\ell -1,L{)}^2{)}^{2|V(H)|},$$

$$\sum _{v_0\in V(G):g(v_0)\ge r}g(v_0)\ge \frac{|S|{\delta }^k}{2}.$$

Full text

1 Introduction

For a multigraph $F$, a subdivision of $F$ is a graph obtained by replacing the edges of $F$ with pairwise internally vertex-disjoint paths of arbitrary lengths. The $k$-subdivision of $F$ is the graph obtained by replacing the edges of $F$ with pairwise internally vertex-disjoint paths of length $k+1$, and is denoted by $F^{k}$.

Many researchers have studied the problem of estimating the number of edges needed in a graph $G$ on $n$ vertices to guarantee that it contains as a subgraph a subdivided copy of a fixed graph. The first result in this direction is due to Mader [12] who proved that for any graph $F$ there exists a constant $c_{F}=c$ such that if an $n$-vertex graph $G$ contains at least $cn$ edges, then $G$ contains a subdivision of $F$ as a subgraph. In this result the size of the subdivided graph can grow with $n$, which is necessary since an $n$-vertex graph with $cn$ edges need not contain a cycle of bounded length.

Answering a question of Erdős about planar subgraphs [5], Kostochka and Pyber [11] proved that any $n$-vertex graph with at least $4^{t^{2}}n^{1+\varepsilon}$ edges contains a subdivided $K_{t}$ with at most $\frac{7t^{2}\log t}{\varepsilon}$ vertices. This is the first result that guarantees a subdivided $K_{t}$ of bounded size.

For a family $\mathcal{F}$ of graphs, we let $\mathrm{ex}(n,\mathcal{F})$ be the maximum number of edges in an $n$-vertex graph not containing any $F\in\mathcal{F}$ as a subgraph. When $\mathcal{F}=\{F\}$, we write $\mathrm{ex}(n,F)$ for the same function.

Let $\mathcal{F}_{t,k}$ be the family of graphs that can be obtained by replacing the edges of $K_{t}$ with pairwise internally vertex-disjoint paths of length at most $k$. Jiang [9] proved that for any $t\in\mathbb{N}$ and any $0<\varepsilon<1/2$, we have $\mathrm{ex}(n,\mathcal{F}_{t,\lceil 10/\varepsilon\rceil})=O(n^{1+\varepsilon})$. Here the asymptotic notation means that $n\rightarrow\infty$ and other parameters are constant. We follow the same convention throughout the paper.

Note that Jiang’s result improves that of Kostochka and Pyber in two ways. Firstly, any $F\in\mathcal{F}_{t,\lceil 10/\varepsilon\rceil}$ has at most $\frac{ct^{2}}{\varepsilon}$ vertices, so a $\log$ factor is saved. Secondly, the edges in Jiang’s theorem are replaced by uniformly short paths not depending on $t$. However, they can still have different lengths. The next result of Jiang and Seiver guarantees a subdivided $K_{t}$ with prescribed path lengths.

Theorem 1.1** (Jiang–Seiver [10]).**

For any $t\in\mathbb{N}$ and any even $k\in\mathbb{N}$,

[TABLE]

Note that if $k$ is odd, then $K_{t}^{k-1}$ is not a bipartite graph, so $\mathrm{ex}(n,K_{t}^{k-1})=\Theta(n^{2})$.

Conlon and Lee conjectured that the following two strengthenings hold.

Conjecture 1.2** (Conlon–Lee [4]).**

Let $F$ be a multigraph and let $k\geq 2$ be even. Then

[TABLE]

Conjecture 1.3** (Conlon–Lee [4]).**

Let $F$ be a simple graph and let $k\geq 2$ be even. Then there exists some $\varepsilon>0$ such that

[TABLE]

In the case $k=2$, Conjecture 1.2 follows from the $r=2$ case of a result of Füredi [7] and Alon, Krivelevich and Sudakov [1], which states that any bipartite graph with maximum degree at most $r$ on one side has extremal number $O(n^{2-1/r})$. The $k=2$ case of Conjecture 1.3 was proved by Conlon and Lee [4], and improved bounds were given by the author [8].

Very recently, Conlon, Janzer and Lee proved Conjecture 1.3 for every bipartite graph $F$.

Theorem 1.4** (Conlon–Janzer–Lee [3]).**

Let $F$ be a simple bipartite graph and let $k\geq 1$. Then there exists some $\varepsilon>0$ such that

[TABLE]

As a simple corollary, they significantly improved the bound in Theorem 1.1.

Theorem 1.5** (Conlon–Janzer–Lee [3]).**

Let $F$ be a simple graph and let $k\geq 2$ be even. Then there exists some $\varepsilon>0$ such that

[TABLE]

In this paper, we prove both Conjecture 1.2 and Conjecture 1.3.

Theorem 1.6**.**

Let $F$ be a multigraph and let $k\geq 2$ be even. Then

[TABLE]

Theorem 1.7**.**

Let $F$ be a simple graph and let $k\geq 2$ be even. Then there exists some $\varepsilon>0$ such that

[TABLE]

Note that these results are tight. Indeed, by a result of Conlon [2], the Theta graph $\theta_{k,\ell}$ has extremal number $\Theta(n^{1+1/k})$ for all $\ell\geq\ell_{0}(k)$, showing that Theorem 1.6 is tight. Moreover, Erdős-Rényi random graphs show that $\mathrm{ex}(n,K_{t}^{k-1})=\Omega(n^{1+1/k-c_{k,t}})$ where $c_{k,t}\rightarrow 0$ as $t\rightarrow\infty$, so Theorem 1.7 is also tight.

The rest of the paper is organised as follows. In Section 2, we introduce some of the key definitions and give the high-level structure of the proof, with the key technical lemmas deferred to Sections 3 and 4.

2 The high-level structure of the proof

A graph $G$ is called $K$-almost-regular if $\max_{v\in V(G)}d(v)\leq K\min_{v\in V(G)}d(v)$, where $d(v)$ is the degree of vertex $v$. The following lemma, which is a small modification of a result proved by Erdős and Simonovits [6], allows us to restrict our attention to almost regular host graphs.

Lemma 2.1** (Jiang–Seiver [10]).**

Let $\varepsilon,c$ be positive reals, where $\varepsilon<1$ and $c\geq 1$. Let $n$ be a positive integer that is sufficiently large as a function of $\varepsilon$. Let $G$ be a graph on $n$ vertices with $e(G)\geq cn^{1+\varepsilon}$. Then $G$ contains a $K$-almost-regular subgraph $G^{\prime}$ on $m\geq n^{\frac{\varepsilon-\varepsilon^{2}}{2+2\varepsilon}}$ vertices such that $e(G^{\prime})\geq\frac{2c}{5}m^{1+\varepsilon}$ and $K=20\cdot 2^{\frac{1}{\varepsilon^{2}}+1}$.

Using this lemma, Theorem 1.6 and Theorem 1.7 reduce to the following two statements, respectively. For notational convenience, we have dropped the assumption that $k$ is even, and replaced $k$ by $2k$.

Theorem 2.2**.**

Let $F$ be a multigraph and let $k\geq 1$. Suppose that $G$ is a $K$-almost-regular graph on $n$ vertices with minimum degree $\delta=\omega(n^{\frac{1}{2k}})$. Then, for $n$ sufficiently large, $G$ contains a copy of $F^{2k-1}$.

Theorem 2.3**.**

Let $F$ be a simple graph and let $k\geq 1$. Then there exists $\varepsilon>0$ with the following property. Suppose that $G$ is a $K$-almost-regular graph on $n$ vertices with minimum degree $\delta=\omega(n^{\frac{1}{2k}-\varepsilon})$. Then, for $n$ sufficiently large, $G$ contains a copy of $F^{2k-1}$.

From now on we let $F$ be an arbitrary fixed multigraph and write $H=F^{2k-1}$. Moreover, throughout the paper we tacitly assume that $n$ is sufficiently large.

The next definition was introduced in [3], and was used to prove Theorem 1.4.

Definition 2.4**.**

Let $L$ be a positive real and let $f(\ell,L)=L^{5^{\ell}}$ for $1\leq\ell\leq 2k$. We recursively define the notions of $L$-admissible and $L$-good paths of length $\ell$ in a graph. Any path of length $1$ is both $L$-admissible and $L$-good. For $2\leq\ell\leq 2k$, we say a path $P=v_{0}v_{1}\dots v_{\ell}$ is $L$-admissible if every proper subpath of $P$ is $L$-good, i.e., $v_{i}v_{i+1}\dots v_{j}$ is $L$-good for every $(i,j)\neq(0,\ell)$. The path $P$ is $L$-good if it is $L$-admissible and the number of $L$-admissible paths of length $\ell$ between $v_{0}$ and $v_{\ell}$ is at most $f(\ell,L)$.

The next lemma will be used several times later.

Lemma 2.5**.**

Let $\ell\geq 2$ and let $L>\ell$. If a path $P=v_{0}\dots v_{\ell}$ is $L$-admissible, but not $L$-good, then there exist at least $L$ pairwise internally vertex-disjoint paths of length $\ell$ from $v_{0}$ to $v_{\ell}$.

Proof..

Take a maximal set of pairwise internally vertex-disjoint paths of length $\ell$ from $v_{0}$ to $v_{\ell}$ and assume that it consists of fewer than $L$ paths. These paths contain at most $L(\ell-1)$ internal vertices in total and any path of length $\ell$ between $v_{0}$ and $v_{\ell}$ intersects at least one of these vertices. Since there are at least $L^{5^{\ell}}$ $L$-admissible paths of length $\ell$ between $v_{0}$ and $v_{\ell}$, it follows by pigeon hole that there exist some $1\leq i\leq\ell-1$ and some $x\in V(G)$ such that there are at least $\frac{L^{5^{\ell}}}{(\ell-1)L(\ell-1)}$ $L$-admissible paths of the form $u_{0}u_{1}\dots u_{\ell}$ with $u_{0}=v_{0},u_{i}=x,u_{\ell}=v_{\ell}$. Observe that $\frac{L^{5^{\ell}}}{(\ell-1)L(\ell-1)}>L^{5^{i}}L^{5^{\ell-i}}$, so either there are more than $L^{5^{i}}$ $L$-good paths of length $i$ between $v_{0}$ and $x$ or there are more than $L^{5^{\ell-i}}$ $L$-good paths of length $\ell-i$ between $x$ and $v_{\ell}$. In either case, we contradict the definition of an $L$-good path. ∎

Our strategy will be to prove that, roughly speaking, in any almost regular $H$-free graph there are many good paths of length $2k$. As we will see in Section 3, the techniques in [3] can be easily applied to prove this for paths of length $k$. The novelty of this paper is the machinery that allows us to extend this to longer paths, using very different techniques. This is given in Section 4, where we prove the following lemma.

Lemma 2.6**.**

Let $G$ be an $H$-free $K$-almost-regular graph on $n$ vertices with minimum degree $\delta\geq L^{100^{k}|V(H)|}$, and let $S\subset V(G)$. Then, provided that $L$ is sufficiently large compared to $|V(H)|$ and $K$, $|S|=\omega(\frac{n}{\delta^{1/2}})$ and $|S|=\omega(\frac{n}{L^{1/2}})$, the number of $L$-good paths of length $2k$ with both endpoints in $S$ is $\Omega(\frac{|S|^{2}\delta^{2k}}{n})$.

Note that in this result and everywhere else in the paper, the asymptotic notation $\Omega$ allows the implied constant to depend on $k,|V(H)|$ and $K$, which are thought of as constants, while $\delta$ and $L$ are functions of $n$.

With Lemma 2.6 in hand, the proof of Theorem 2.2 is immediate.

Proof of Theorem 2.2.

Suppose that $G$ does not contain $H=F^{2k-1}$ as a subgraph. Since $\delta=\omega(n^{\frac{1}{2k}})$, we may choose $L$ with $L=\omega(1)$, $L^{100^{k}|V(H)|}\leq\delta$ and $n^{2}f(2k,L)=o(n\delta^{2k})$. Then we may apply Lemma 2.6 with $S=V(G)$ to get that the number of $L$-good paths of length $2k$ in $G$ is $\Omega(n\delta^{2k})$, which is $\omega(n^{2}f(2k,L))$. However, by the definition of $L$-goodness, between any two vertices there can be at most $f(2k,L)$ such paths, which is a contradiction. ∎

The proof of Theorem 2.3 is slightly more complicated, and it uses ideas from [8].

Proof of Theorem 2.3.

Firstly note that $F$ is a subgraph of $K_{t}$ for some $t$, so it suffices to prove the result for $F=K_{t}$. Let $\varepsilon>0$ be sufficiently small, to be specified, and let $G$ be a $K$-almost-regular graph on $n$ vertices with minimum degree $\delta=\omega(n^{\frac{1}{2k}-\varepsilon})$. Assume that $G$ does not contain a copy of $H=F^{2k-1}$.

For vertices $u,v\in V(G)$, let us write $u\sim v$ if there is a path of length $2k$ between $u$ and $v$. Also, let us say that $u$ and $v$ are distant if for every $1\leq i\leq 4k-2$, the number of walks of length $i$ between $u$ and $v$ is at most $\delta^{i-2k+1/2}$. Observe that for any $u\in V(G)$ the number of walks of length $i$ starting from $u$ is at most $(K\delta)^{i}$, so the number of vertices $v\in V(G)$ for which there are at least $\delta^{i-2k+1/2}$ walks of length $i$ from $u$ to $v$ is at most $\frac{(K\delta)^{i}}{\delta^{i-2k+1/2}}=K^{i}\delta^{2k-1/2}$. Thus, the number of $v\in V(G)$ for which $u$ and $v$ are not distant is $O(\delta^{2k-1/2})$.

Define $c_{0}=\varepsilon$ and $c_{\ell+1}=(3\cdot 5^{2k}+1)c_{\ell}+2k\varepsilon$ for $0\leq\ell\leq t-1$. Assume that $\varepsilon$ is small enough so that

[TABLE]

for all $0\leq\ell\leq t$. Then in particular $c_{\ell}\leq\frac{1}{4k}-\varepsilon/2$ holds for all $0\leq\ell\leq t$. For future reference, note that then

[TABLE]

Claim. For any $0\leq\ell\leq t$, there exist distinct vertices $x_{1},\dots,x_{\ell}\in V(G)$ and a set $S_{\ell}\subset V(G)$ such that

1. (i)

there is a copy of $K_{\ell}^{2k-1}$ in $G$ with the vertices of the subdivided $K_{\ell}$ being $x_{1},\dots,x_{\ell}$ 2. (ii)

$x_{i}\sim y$ for every $1\leq i\leq\ell$ and every $y\in S_{\ell}$ 3. (iii)

$|S_{\ell}|=\Omega(n^{1-c_{\ell}})$ and 4. (iv)

$x_{i}$ and $x_{j}$ are distant for every $1\leq i<j\leq\ell$.

Note that in particular for $\ell=t$, condition (i) guarantees the existence of a subgraph $K_{t}^{2k-1}$, so it suffices to prove the claim.

Proof of Claim. We proceed by induction on $\ell$. For $\ell=0$, we may take $S_{0}=V(G)$. Assume now that we have verified the claim for $\ell$.

Suppose that for some $y\in S_{\ell}$ there exist $1\leq i<j\leq\ell$ and two paths of length $2k$, one (called $P_{i}$) from $x_{i}$ to $y$ and one (called $P_{j}$) from $x_{j}$ to $y$, which share a vertex other than $y$. Let they intersect at some vertex $z\neq y$. Now let the subpath of $P_{i}$ between $x_{i}$ and $z$ have length $\alpha$ and let the subpath of $P_{j}$ between $x_{j}$ and $z$ have length $\beta$. Then there is a walk of length $\alpha+\beta$ from $x_{i}$ to $x_{j}$ through $z$. Moreover, there is a path of length $2k-\alpha$ from $z$ to $y$. Observe that $2k-\alpha\leq 4k-(\alpha+\beta)-1$.

Let $Y$ be the set of $y\in S_{\ell}$ for which there exist some $1\leq i<j\leq\ell$ and a walk $W$ of length $\gamma\leq 4k-2$ between $x_{i}$ and $x_{j}$ such that for some vertex $w$ on $W$ the distance of $y$ from $w$ is at most $4k-\gamma-1$. By condition (iv), there are at most $\delta^{\gamma-2k+1/2}$ walks of length $\gamma$ between any $x_{i}$ and $x_{j}$ so there are $O(\delta^{\gamma-2k+1/2})$ vertices appearing in at least one of these walks. Therefore the number of vertices at distance at most $4k-\gamma-1$ from at least one of these vertices is $O(\delta^{\gamma-2k+1/2}\cdot\delta^{4k-\gamma-1})=O(\delta^{2k-1/2})$. That is, $|Y|=O(\delta^{2k-1/2})$.

Notice that by the discussion above, for any $y\in S_{\ell}\setminus Y$ and any $i\neq j$, a path of length $2k$ from $x_{i}$ to $y$, and a path of length $2k$ from $x_{j}$ to $y$ have no common vertex other than $y$. Thus, by condition (ii) there exist $\ell$ paths of length $2k$, one from each $x_{i}$ to $y$ which are pairwise vertex-disjoint apart from at $y$. Moreover, these paths are also vertex-disjoint from the paths forming the $K_{\ell}^{2k-1}$ guaranteed by condition (i), apart from the trivial intersections at $x_{1},\dots,x_{\ell}$ (else, there is a path of length at most $2k-1$ from $y$ to a point on a path of length $2k$ between some $x_{i}$ and $x_{j}$, which contradicts the fact that $y\not\in Y$). Thus, for any $y\in S_{\ell}\setminus Y$ there is a copy of $K_{\ell+1}^{2k-1}$ in $G$ with the vertices of the subdivided $K_{\ell+1}$ being $x_{1},\dots,x_{\ell},y$.

Let $Z$ be the set of $z\in S_{\ell}$ which are not distant to $x_{i}$ for at least one $1\leq i\leq\ell$. By the second paragraph in this proof, $|Z|=O(\delta^{2k-1/2})$.

Let $S^{\prime}_{\ell}=S_{\ell}\setminus(Y\cup Z)$. Recall that $|Y|=O(\delta^{2k-1/2})$. Note that if $\delta=\omega(n^{\frac{1}{2k}})$, then, by Theorem 2.2, $G$ contains $H$ as a subgraph, so we may assume that $\delta=O(n^{\frac{1}{2k}})$. Then $\delta^{2k-1/2}=O(\frac{n}{\delta^{1/2}})$, which is $o(n^{1-c_{\ell}})$ by equation (2). Thus, $|Y\cup Z|=o(n^{1-c_{\ell}})$ and so $|S^{\prime}_{\ell}|=\Omega(n^{1-c_{\ell}})$.

Let $L=n^{3c_{\ell}}$. Then, by equation (1), we have $L^{100^{k}|V(H)|}\leq n^{\frac{1}{2k}-\varepsilon}=o(\delta)$. Moreover, by equation (2), we have $n^{1-c_{\ell}}=\omega(\frac{n}{\delta^{1/2}})$, and by the definition of $L$, we have $n^{1-c_{\ell}}=\omega(\frac{n}{L^{1/2}})$. Hence, by Lemma 2.6, the number of $L$-good paths of length $2k$ with both endpoints in $S^{\prime}_{\ell}$ is $\Omega(\frac{|S^{\prime}_{\ell}|^{2}\delta^{2k}}{n})$. Between any two vertices in $S^{\prime}_{\ell}$ there are at most $f(2k,L)$ $L$-good paths of length $2k$, so the number of pairs $(z,y)\in S^{\prime}_{\ell}\times S^{\prime}_{\ell}$ with $z\sim y$ is $\Omega(\frac{|S^{\prime}_{\ell}|^{2}\delta^{2k}}{nf(2k,L)})$. Thus, there exists some $x_{\ell+1}\in S^{\prime}_{\ell}$ such that the number of $y\in S^{\prime}_{\ell}$ with $x_{\ell+1}\sim y$ is $\Omega(\frac{|S^{\prime}_{\ell}|\delta^{2k}}{nf(2k,L)})\geq\Omega(\frac{n^{1-c_{\ell}-2k\varepsilon}}{L^{5^{2k}}})=\Omega(n^{1-c_{\ell}-2k\varepsilon-3c_{\ell}5^{2k}})=\Omega(n^{1-c_{\ell+1}})$. Set $S_{\ell+1}$ to be the set of these $y\in S^{\prime}_{\ell}$, and note that properties (i)-(iv) are satisfied for $\ell+1$. ∎

3 Short paths

Our aim in this section is to prove the following lemma.

Lemma 3.1**.**

Let $G$ be an $H$-free $K$-almost-regular graph on $n$ vertices with minimum degree $\delta\geq L^{100^{k}|V(H)|}$. Then, provided that $L$ is sufficiently large compared to $|V(H)|$ and $K$, the number of paths of length $k$ that are not good is $O(\frac{n\delta^{k}}{L})$.

The proof of this is almost identical to that of Lemma 6.4 in [3], nevertheless we include it here for completeness and since some minor details need to be modified.

The next definition is for notational convenience.

Definition 3.2**.**

A pair of distinct vertices $(x,y)$ in $G$ is said to be $(\ell,L)$-bad for some $2\leq\ell\leq 2k$ and some $L$ if there is an $L$-admissible, but not $L$-good, path of length $\ell$ from $x$ to $y$.

In what follows, for $v\in V(G)$, we shall write $\Gamma_{i}(v)$ for the set of vertices $u\in V(G)$ for which there exists a path of length $i$ from $v$ to $u$ and write $N(v)=\Gamma_{1}(v)$. The next lemma is a slight variant of Lemma 6.7 from [3].

Lemma 3.3**.**

Let $2\leq\ell\leq k$ and $1\leq i\leq\ell$. Let $G$ be a $K$-almost-regular graph on $n$ vertices with minimum degree $\delta>0$. Let $X,Y,Z\subset V(G)$ be such that $|Z|\leq L^{1/10},|Y|\leq(K\delta)^{\ell-1}$ and, for any $x\in X$, the number of $y\in Y$ such that $(x,y)$ is $(\ell,L)$-bad is as at least $\frac{(K\delta)^{\ell-1}}{f(\ell-1,L)^{2}}$. Then, provided that $L$ is sufficiently large compared to $k$ and $K$, there exist a path of length $2i$ in $G$, disjoint from $Z$, whose endpoints form a set $R\subset Y$, and a subset $X^{\prime}\subset X$ such that $|X^{\prime}|\geq|X\setminus Z|/(16f(\ell-1,L)^{2})$ and $(x^{\prime},r)$ is $(\ell,L)$-bad for every $x^{\prime}\in X^{\prime}$ and $r\in R$.

Proof..

After replacing $X$ by $X\setminus Z$, we may assume $X\cap Z=\emptyset$. Let $Y^{\prime}$ be the set of those $y\in Y$ for which the number of $x\in X$ such that $(x,y)$ is $(\ell,L)$-bad is at least $\frac{|X|}{2f(\ell-1,L)^{2}}$. Then the number of $(x,y)\in X\times(Y\setminus Y^{\prime})$ which are $(\ell,L)$-bad is at most $\frac{|X||Y|}{2f(\ell-1,L)^{2}}\leq\frac{|X|(K\delta)^{\ell-1}}{2f(\ell-1,L)^{2}}$, so the number of $(x,y)\in X\times Y^{\prime}$ which are $(\ell,L)$-bad is at least $\frac{|X|(K\delta)^{\ell-1}}{2f(\ell-1,L)^{2}}$. Now there exists some $x^{*}\in X$ such that there are at least $\frac{(K\delta)^{\ell-1}}{2f(\ell-1,L)^{2}}$ choices $y\in Y^{\prime}$ for which $(x^{*},y)$ is $(\ell,L)$-bad. If a pair $(x^{*},y)$ is $(\ell,L)$-bad, then there are at least $f(\ell,L)$ paths of length $\ell$ from $x^{*}$ to $y$. Hence, there are at least $\frac{(K\delta)^{\ell-1}}{2f(\ell-1,L)^{2}}\cdot f(\ell,L)=\Omega(f(\ell-1,L)^{3}\delta^{\ell-1})$ paths of length $\ell$ starting at $x^{*}$ and ending in $Y^{\prime}$.

The number of such paths intersecting $Z$ is at most $|Z|\ell(K\delta)^{\ell-1}$. Indeed, there are at most $|Z|$ choices for the element of $Z$ in the path, at most $\ell$ choices for its position in the path and, given a fixed choice for these, at most $(K\delta)^{\ell-1}$ choices for the other $\ell-1$ vertices in the path. (Note that as $X\cap Z=\emptyset$, the vertex in $Z$ is not $x^{*}$.) But $|Z|\ell(K\delta)^{\ell-1}\leq L^{1/10}\ell K^{\ell-1}\delta^{\ell-1}$, so, for $L$ sufficiently large there are $\Omega(f(\ell-1,L)^{3}\delta^{\ell-1})$ paths of length $\ell$ starting at $x^{*}$ and ending in $Y^{\prime}$ that avoid $Z$. Moreover, since $|\Gamma_{\ell-i}(x^{*})|\leq(K\delta)^{\ell-i}$, it follows that there exists some $u\in\Gamma_{\ell-i}(x^{*})$ such that there are $\Omega(f(\ell-1,L)^{3}\delta^{i-1})$ paths of length $i$ from $u$ to $Y^{\prime}$, all avoiding $Z$.

Take now a maximal set of such paths which are pairwise vertex-disjoint apart from at $u$. We claim that there are $\Omega(f(\ell-1,L)^{3})$ such paths. Suppose otherwise. Then all the $\Omega(f(\ell-1,L)^{3}\delta^{i-1})$ paths of length $i$ from $u$ to $Y^{\prime}$ intersect a certain set of size $o(f(\ell-1,L)^{3})$ not containing $u$. But there are $o(f(\ell-1,L)^{3})\delta^{i-1}$ such paths, which is a contradiction.

So we have $r=\Omega(f(\ell-1,L)^{3})$ paths $P_{1},\dots,P_{r}$ of length $i$ from $u$ to $Y^{\prime}$ which are pairwise vertex-disjoint except at $u$ and avoid $Z$. Let the endpoints of these paths be $y_{1},\dots,y_{r}$. Since $y_{j}\in Y^{\prime}$ for all $j$, the number of pairs $(x,y_{j})$ with $x\in X$ which are $(\ell,L)$-bad is at least $\frac{r|X|}{2f(\ell-1,L)^{2}}$. Therefore, by Jensen’s inequality, for an average $x\in X$ there are at least ${r/(2f(\ell-1,L)^{2})\choose 2}$ choices $1\leq j_{1}<j_{2}\leq r$ such that both $(x,y_{j_{1}})$ and $(x,y_{j_{2}})$ are $(\ell,L)$-bad. Since ${r/(2f(\ell-1,L)^{2})\choose 2}\geq(\frac{1}{4f(\ell-1,L)^{2}})^{2}{r\choose 2}$, there exist $1\leq j_{1}<j_{2}\leq r$ such that the set

[TABLE]

has size at least $|X|/(4f(\ell-1,L)^{2})^{2}$. We can now take $R=\{y_{j_{1}},y_{j_{2}}\}$, and the union of the paths $P_{j_{1}}$ and $P_{j_{2}}$ is a suitable path of length $2i$. ∎

The following lemma is a small modification of Lemma 6.8 from [3].

Lemma 3.4**.**

Let $G$ be an $H$-free $K$-almost-regular graph on $n$ vertices with minimum degree $\delta\geq L^{100^{k}|V(H)|}$. Let $2\leq\ell\leq k$ and any $v\in V(G)$. Then, provided that $L$ is sufficiently large compared to $|V(H)|$ and $K$, the number of $L$-admissible, but not $L$-good, paths of the form $v_{0}vv_{2}v_{3}\dots v_{\ell}$ is at most $\frac{2(K\delta)^{\ell}}{f(\ell-1,L)}$.

Proof..

Suppose otherwise. Let $Y=\Gamma_{\ell-1}(v)$ and note that $|Y|\leq(K\delta)^{\ell-1}$. For any $x\in N(v)$ and any $y\in Y$, the number of $L$-admissible paths of the form $xvv_{2}\dots v_{\ell-1}y$ is at most $f(\ell-1,L)$. Indeed, in any such path, the subpath $vv_{2}v_{3}\dots v_{\ell-1}y$ is $L$-good, and for any fixed $y\in Y$ there are at most $f(\ell-1,L)$ such $L$-good paths. Hence, by assumption, the number of pairs $(x,y)\in N(v)\times Y$ such that there is an $L$-admissible, but not $L$-good, path of the form $xvv_{1}\dots v_{\ell-1}y$ is at least $\frac{2(K\delta)^{\ell}}{f(\ell-1,L)^{2}}\geq\frac{2|N(v)|(K\delta)^{l-1}}{f(\ell-1,L)^{2}}$. By definition, any such pair $(x,y)$ is $(\ell,L)$-bad. Let $X$ consist of those $x\in N(v)$ for which there are at least $\frac{(K\delta)^{\ell-1}}{f(\ell-1,L)^{2}}$ choices of $y\in Y$ such that $(x,y)$ is $(\ell,L)$-bad. Then the number of pairs $(x,y)\in X\times Y$ which are $(\ell,L)$-bad is at least $\frac{|N(v)|(K\delta)^{\ell-1}}{f(\ell-1,L)^{2}}$, and so $|X|\geq\frac{|N(v)|}{f(\ell-1,L)^{2}}\geq\frac{\delta}{f(\ell-1,L)^{2}}$.

Our aim now is to find a copy of $H$ in $G$, which will yield a contradiction. Write $k=j\ell+i$ with $1\leq i\leq\ell$.

Note that if $L$ is sufficiently large, then

[TABLE]

so we may apply Lemma 3.3 repeatedly $|E(F)|+|V(H)|\leq 2|V(H)|$ times and still get a set $X^{\prime}$ of size at least $L$. Thus, we find disjoint paths $P_{e}$ of length $2i$ for every $e\in E(F)$ whose endpoint sets are $R_{e}\subset Y$, and sets $X_{\text{final}}\subset X$ and $U\subset Y$ with $|X_{\text{final}}|=|U|=|V(H)|$ such that $V(P_{e}),X_{\text{final}}$ and $U$ are pairwise disjoint and any pair $(x,y)$ with $x\in X_{\text{final}}$ and $y\in U\cup\bigcup_{e\in E(F)}R_{e}$ is $(\ell,L)$-bad. For $e\in E(F)$, let $y_{e_{-k}}y_{e_{-k+1}}\dots y_{e_{k}}$ be the path of length $2k$ replacing the edge $e$.

A copy of $H$ in $G$ can now be constructed as follows. For each $e\in E(F)$, map the path $y_{e_{-i}}y_{e_{-i+1}}\dots y_{e_{i}}$ to $P_{e}$. Then map, for each $e\in E(F)$, the vertices $y_{e_{i+\ell}},y_{e_{-(i+\ell)}}$ to $X_{\text{final}}$ in an arbitrary injective manner. Also, map each $y_{e_{i+2\ell}},y_{e_{-(i+2\ell)}}$ to $U$ in an arbitrary injective manner. More generally, map the vertices $y_{e_{i+a\ell}},y_{e_{-(i+a\ell)}}$ with $a\geq 1$ odd to $X_{\text{final}}$ in an arbitrary injective manner and map the vertices $y_{e_{i+a\ell}},y_{e_{-(i+a\ell)}}$ with $a\geq 2$ even to $U$ in an arbitrary injective manner. We then just need to find paths of length $\ell$ connecting $y_{e_{i+a\ell}}$ and $y_{e_{i+(a+1)\ell}}$ (and paths of length $\ell$ connecting $y_{e_{-(i+a\ell)}}$ and $y_{e_{-(i+(a+1)\ell)}}$) which are disjoint from each other and from the images of the already mapped vertices. Since $(x,y)$ is $(\ell,L)$-bad for every $x\in X_{\text{final}}$ and $y\in U\cup\bigcup_{e\in E(F)}R_{e}$, such paths exist by Lemma 2.5, provided that $L$ is sufficiently large. ∎

Corollary 3.5**.**

Let $G$ be an $H$-free $K$-almost-regular graph on $n$ vertices with minimum degree $\delta\geq L^{100^{k}|V(H)|}$. Then, provided that $L$ is sufficiently large compared to $|V(H)|$ and $K$, for any $2\leq\ell\leq k$, the number of $L$-admissible, but not $L$-good, paths of length $\ell$ is at most $n\frac{2(K\delta)^{\ell}}{f(\ell-1,L)}$.

Now we are in a position to prove Lemma 3.1.

Proof of Lemma 3.1.

Suppose that the path $u_{0}u_{1}\dots u_{k}$ is not $L$-good. Take $0\leq i<j\leq k$ with $j-i$ minimal such that $u_{i}u_{i+1}\dots u_{j}$ is not $L$-good. Then $u_{i}\dots u_{j}$ is $L$-admissible. For any fixed $i,j$, by Corollary 3.5, the number of such paths is at most $n\frac{2(K\delta)^{j-i}}{f(j-i-1,L)}\cdot 2(K\delta)^{k-(j-i)}=4K^{k}\frac{n\delta^{k}}{f(j-i-1,L)}\leq 4K^{k}\frac{n\delta^{k}}{L}$. Using that $i$ and $j$ can take at most $k+1$ values each, it follows that the number of not $L$-good paths of length $k$ is at most $(k+1)^{2}4K^{k}\cdot\frac{n\delta^{k}}{L}$. ∎

4 Long paths

In what follows, for a vertex $x\in V(G)$ and a nonnegative integer $i$, we write $\mathcal{P}_{i}(x)$ for the set of directed paths of length $i$ starting at $x$. For an element $P\in\mathcal{P}_{i}(x)$, we let $v(P)$ be the endpoint of the path $P$.

Definition 4.1**.**

Let $i,j$ be nonnegative integers with $i+j<2k$. Call a pair $(x,y)$ of distinct vertices $(i,j)$-rich if the number of pairs $(P,Q)\in\mathcal{P}_{i}(x)\times\mathcal{P}_{j}(y)$ such that there are at least $(|V(H)|+2)(2k+1)+1$ pairwise internally vertex-disjoint paths of length $2k-i-j$ between $v(P)$ and $v(Q)$ is more than $(2(i+j)|V(H)|(2k+1)+2(i+1)j)(K\delta)^{i+j-1}$. Otherwise (including when $x=y$) call it $(i,j)$-poor.

Lemma 4.2**.**

Let $G$ be a graph with maximum degree at most $K\delta$. Let $x,y\in V(G)$ and let $i,j$ be nonnegative integers with $i+j<2k$. If $(x,y)$ is $(i,j)$-rich, then there exist $|V(H)|$ pairwise internally vertex-disjoint paths of length $2k$ between $x$ and $y$.

Proof..

Choose a maximal set of pairwise internally vertex-disjoint paths $R_{1},\dots,R_{\alpha}$ between $x$ and $y$ and assume that $\alpha<|V(H)|$. Let $T$ be the set of the vertices appearing in at least one of these paths. Note that $|T|<|V(H)|(2k+1)$.

Claim. If there is a pair $(P,Q)\in\mathcal{P}_{i}(x)\times\mathcal{P}_{j}(y)$ such that

1. (i)

$P$ is disjoint from $T\setminus\{x\}$ 2. (ii)

$Q$ is disjoint from $T\setminus\{y\}$ 3. (iii)

$P$ and $Q$ are vertex-disjoint and 4. (iv)

there are at least $(|V(H)|+2)(2k+1)+1$ pairwise internally vertex-disjoint paths of length $2k-i-j$ between $v(P)$ and $v(Q)$,

then there is a path of length $2k$ between $x$ and $y$ which is internally vertex-disjoint from all of $R_{1},\dots,R_{\alpha}$.

Proof of Claim. Clearly, it suffices to find a path of length $2k-i-j$ between $v(P)$ and $v(Q)$ which is disjoint from the vertices of $R_{1},\dots,R_{\alpha},P,Q$, except for $v(P)$ and $v(Q)$. But such a path exists since there are at most $(\alpha+2)\cdot(2k+1)\leq(|V(H)|+2)(2k+1)$ vertices in one of $R_{1},\dots,R_{\alpha},P,Q$ and there are at least $(|V(H)|+2)(2k+1)+1$ pairwise internally vertex-disjoint paths of length $2k-i-j$ between $v(P)$ and $v(Q)$.

A path provided by the claim would contradict the maximality of $R_{1},\dots,R_{\alpha}$, so it suffices to prove that there are paths $P,Q$ satisfying (i)-(iv) above.

Since the maximum degree of $G$ is at most $K\delta$, the number of paths of length $i-1$ in $G$ intersecting $T$ is at most $i|T|(K\delta)^{i-1}$, so the number of $P\in\mathcal{P}_{i}(x)$ which have a vertex in $T\setminus\{x\}$ is at most $2i|T|(K\delta)^{i-1}$. Since $|\mathcal{P}_{j}(y)|\leq(K\delta)^{j}$, the number of pairs $(P,Q)\in\mathcal{P}_{i}(x)\times\mathcal{P}_{j}(y)$ failing condition (i) above is at most $2i|T|(K\delta)^{i-1}(K\delta)^{j}$. Similarly, the number of pairs failing (ii) is at most $2j|T|(K\delta)^{j-1}(K\delta)^{i}$. Finally, for every $P\in\mathcal{P}_{i}(x)$, the number of paths of length $j-1$ which intersect $P$ is at most $(i+1)j(K\delta)^{j-1}$, so the number of pairs $(P,Q)\in\mathcal{P}_{i}(x)\times\mathcal{P}_{j}(y)$ for which $P$ and $Q$ share a vertex other than $y$ is at most $(K\delta)^{i}\cdot 2(i+1)j(K\delta)^{j-1}$. So the number of pairs which fail at least one of (i),(ii),(iii) is at most $(2(i+j)|T|+2(i+1)j)(K\delta)^{i+j-1}\leq(2(i+j)|V(H)|(2k+1)+2(i+1)j)(K\delta)^{i+j-1}$. By the definition of $(i,j)$-richness of $(x,y)$ it follows that there is a pair $(P,Q)$ satisfying (i)-(iv). ∎

Definition 4.3**.**

For a vertex $v\in V(G)$ and some $1\leq\ell\leq k$, define an auxiliary graph $\mathcal{G}_{\ell}(v)$ as follows. The vertices of $\mathcal{G}_{\ell}(v)$ are the $(k+1)$-tuples $(u_{0},u_{1},\dots,u_{k})\in V(G)^{k+1}$ with $u_{0}=v$ such that $u_{i}u_{i+1}\in E(G)$ for all $i$. Vertices $(u_{0},\dots,u_{k})$ and $(u^{\prime}_{0},\dots,u^{\prime}_{k})$ are joined by an edge if $v,u_{1},u_{2},\dots,u_{k},u^{\prime}_{1},\dots,u^{\prime}_{k}$ are distinct and there exist $0\leq i,j\leq k-1$ such that the pair $(u_{\ell},u^{\prime}_{\ell})$ is $(i,j)$-rich. Since the vertex set of $\mathcal{G}_{\ell}(v)$ does not depend on $\ell$, we may define $\mathcal{G}(v)$ to be the union $\bigcup_{1\leq\ell\leq k}\mathcal{G}_{\ell}(v)$.

Lemma 4.4**.**

Let $G$ be a graph with maximum degree at most $K\delta$ which does not contain $H$ as a subgraph. Let $t=|V(F)|$. Then for any $v\in V(G)$ and any $1\leq\ell\leq k$, the graph $\mathcal{G}_{\ell}(v)$ is $K_{t}$-free.

Moreover, let $r=R_{k}(t)$ be the $k$-colour Ramsey number. Then $\mathcal{G}(v)$ is $K_{r}$-free.

Proof..

Suppose that $\mathcal{G}_{\ell}(v)$ contains $K_{t}$ as a subgraph. Let the corresponding vertices be the vectors $u^{1},\dots,u^{t}$. Let their respective $(\ell+1)$th coordinate be $u^{1}_{\ell},\dots,u^{t}_{\ell}$. For every $a\neq b$, since $u^{a}u^{b}$ is an edge in $\mathcal{G}_{\ell}(v)$, it follows that $u^{a}_{\ell}$ and $u^{b}_{\ell}$ are distinct, and, by Lemma 4.2, there exist $|V(H)|$ pairwise internally vertex-disjoint paths of length $2k$ between them. It is not hard to see that this implies that there is a copy of $H$ in $G$ in which the vertices of $F$ are mapped to $u^{1}_{\ell},\dots,u^{t}_{\ell}$. This is a contradiction, so $\mathcal{G}_{\ell}(v)$ is indeed $K_{t}$-free.

Suppose there is a copy of $K_{r}$ in $\mathcal{G}(v)$. Then each edge in this $K_{r}$ can be coloured with one of the colours $1,2,\dots,k$ such that if an edge gets colour $i$, then it lies in $\mathcal{G}_{i}(v)$. By the definition of $r$, there exists a monochromatic $K_{t}$ in this $k$-edge-coloured $K_{r}$, which gives a $K_{t}$ in some $\mathcal{G}_{\ell}(v)$, contradicting the first paragraph. ∎

The next lemma provides us a large set of walks of length $2k$ with both endpoints in $S$. Later, we will argue that most of them are $L$-good paths.

Lemma 4.5**.**

Let $r=R_{k}(t)$ denote the $k$-colour Ramsey number where $t=|V(F)|$. Let $G$ be an $H$-free $K$-almost-regular graph on $n$ vertices with minimum degree $\delta$ and let $S\subset V(G)$ such that $|S|\geq 2nr/\delta^{k}$. Then there are at least $\frac{|S|^{2}\delta^{2k}}{4r^{2}n}$ vectors $(u_{-k},\dots,u_{k})\in V(G)^{2k+1}$ with the following properties

1. (i)

$u_{-k}\in S$, $u_{k}\in S$ 2. (ii)

$u_{\ell}u_{\ell+1}\in E(G)$* for every $-k\leq\ell\leq k-1$* 3. (iii)

$(u_{-\ell},u_{\ell})$* is $(i,j)$-poor for every $1\leq\ell\leq k$ and every $0\leq i,j\leq k-1$.*

Proof..

Since the minimum degree of $G$ is $\delta$, the number of $(k+1)$-tuples $(v_{0},v_{1},\dots,v_{k})\in V(G)^{k+1}$ with $v_{k}\in S$ and $v_{i}v_{i+1}\in E(G)$ for every $0\leq i\leq k-1$ is at least $|S|\delta^{k}$. Writing $\mathcal{T}(v_{0})$ for the set of such vectors for a fixed $v_{0}$ and letting $g(v_{0})=|\mathcal{T}(v_{0})|$, we get that $\sum_{v_{0}\in V(G)}g(v_{0})\geq|S|\delta^{k}$. Note that $\sum_{v_{0}\in V(G):g(v_{0})<r}g(v_{0})\leq nr\leq\frac{|S|\delta^{k}}{2}$, so

[TABLE]

Note that $\mathcal{T}(v_{0})\subset V(\mathcal{G}(v_{0}))$. By Lemma 4.4, the graph $\mathcal{G}(v_{0})[\mathcal{T}(v_{0})]$ is $K_{r}$-free. This graph has $g(v_{0})$ vertices, so if $g(v_{0})\geq r$, then the number of non-edges in $\mathcal{G}(v_{0})[\mathcal{T}(v_{0})]$ is at least $\frac{1}{{r\choose 2}}{g(v_{0})\choose 2}\geq\frac{g(v_{0})^{2}}{r^{2}}$. But if $v=(v_{0},v_{1},\dots,v_{k})\in\mathcal{T}(v_{0})$ and $v^{\prime}=(v_{0},v^{\prime}_{1},\dots,v^{\prime}_{k})\in\mathcal{T}(v_{0})$ are such that $vv^{\prime}$ is not an edge in $\mathcal{G}(v_{0})$, then $(u_{-k},\dots,u_{k})=(v^{\prime}_{k},v^{\prime}_{k-1},\dots,v^{\prime}_{1},v_{0},v_{1},\dots,v_{k})$ satisfies all three properties in the statement of the lemma. Therefore the number of such $(2k+1)$-tuples with $u_{0}=v_{0}$ is at least $\frac{g(v_{0})^{2}}{r^{2}}$ provided that $g(v_{0})\geq r$. By (3) and Jensen’s inequality, we get $\sum_{v_{0}\in V(G):g(v_{0})\geq r}\frac{g(v_{0})^{2}}{r^{2}}\geq\frac{|S|^{2}\delta^{2k}}{4r^{2}n}$, and the proof is complete. ∎

The following simple lemma shows that most walks of length $2k$ are paths.

Lemma 4.6**.**

Let $G$ be a graph on $n$ vertices with maximum degree at most $K\delta$. Then the number of $(2k+1)$-tuples $(u_{-k},\dots,u_{k})\in V(G)^{2k+1}$ such that $u_{i}u_{i+1}\in E(G)$ for every $i$ and $u_{i}=u_{j}$ for some $i\neq j$ is at most ${2k+1\choose 2}K^{2k-1}\cdot n\delta^{2k-1}$.

Proof..

There are ${2k+1\choose 2}$ ways to choose the pair $\{i,j\}$ and there are $n$ ways to choose $u_{i}=u_{j}$. Given any such choices, there are at most $(K\delta)^{2k-1}$ ways to choose the vertices $u_{b}$ for $b\not\in\{i,j\}$ since any vertex in $G$ has degree at most $K\delta$. ∎

Our strategy now is to take all the paths guaranteed by Lemmas 4.5 and 4.6 and discard those which contain a subpath of length $k$ which is not $L$-good. The next result shows that doing this we discard only a small proportion of the paths.

Lemma 4.7**.**

Let $G$ be an $H$-free $K$-almost-regular graph on $n$ vertices with minimum degree $\delta\geq L^{100^{k}|V(H)|}$. Then, provided that $L$ is sufficiently large compared to $|V(H)|$ and $K$, the number of paths $u_{-k}u_{-k+1}\dots u_{k}$ of length $2k$ in $G$ with the property that there is some $-k\leq j\leq 0$ for which the path $u_{j}u_{j+1}\dots u_{j+k}$ is not $L$-good is $O(\frac{n\delta^{2k}}{L})$.

Proof..

By Lemma 3.1, there are $O(\frac{n\delta^{k}}{L})$ paths $u_{j}u_{j+1}\dots u_{j+k}$ which are not $L$-good, and since the maximum degree of $G$ is at most $K\delta$, there are at most $2(K\delta)^{k}$ ways to extend such a path to a path $u_{-k}u_{-k+1}\dots u_{k}$ of length $2k$. The result follows after summing these terms for all $-k\leq j\leq 0$. ∎

The next lemma is the first step to relate the notion of $L$-goodness with the notion of $(i,j)$-richness.

Lemma 4.8**.**

Suppose that $u_{-k}u_{-k+1}\dots u_{k}$ is a path in $G$ which is not $L$-good but each of its subpaths of length $k$ is $L$-good. Then, provided that $L$ is sufficiently large compared to $|V(H)|$, there exist $1\leq\alpha,\beta\leq k$ with $\alpha+\beta>k$ such that there exist $(|V(H)|+2)(2k+1)+1$ pairwise internally vertex-disjoint paths of length $\alpha+\beta$ between $u_{-\alpha}$ and $u_{\beta}$.

Proof..

Choose $-k\leq i<j\leq k$ with $j-i$ minimal such that $u_{i}u_{i+1}\dots u_{j}$ is not $L$-good. By the minimality of $j-i$, every proper subpath of $u_{i}u_{i+1}\dots u_{j}$ is $L$-good, so $u_{i}u_{i+1}\dots u_{j}$ is $L$-admissible. By Lemma 2.5, there exist $(|V(H)|+2)(2k+1)+1$ pairwise internally vertex-disjoint paths of length $j-i$ between $u_{i}$ and $u_{j}$.

By the assumption that every subpath of $u_{-k}u_{-k+1}\dots u_{k}$ of length $k$ is good, we have $j-i>k$, so $i<0$ and $j>0$. Thus, the choices $\alpha=-i$ and $\beta=j$ satisfy the conditions described in the lemma. ∎

The next result is the final ingredient to the proof of Lemma 2.6.

Lemma 4.9**.**

Let $G$ be a graph on $n$ vertices with maximum degree at most $K\delta$. Then there are $O(n\delta^{2k-1})$ paths $u_{-k}u_{-k+1}\dots u_{k}$ in $G$ with the following two properties

1. (i)

$(u_{-\ell},u_{\ell})$* is $(i,j)$-poor for every $1\leq\ell\leq k$ and every $0\leq i,j\leq k-1$ and* 2. (ii)

there exist $1\leq\alpha,\beta\leq k$ with $\alpha+\beta>k$ such that there exist $(|V(H)|+2)(2k+1)+1$ pairwise internally vertex-disjoint paths of length $\alpha+\beta$ between $u_{-\alpha}$ and $u_{\beta}$.

Proof..

Fix a pair $(\alpha,\beta)$ with $1\leq\alpha,\beta\leq k$ and $\alpha+\beta>k$. It suffices to prove that the number of paths satisfying (i) and (ii) for this pair $(\alpha,\beta)$ is $O(n\delta^{2k-1})$.

Let $\ell=\alpha+\beta-k$. Note that $1\leq\ell\leq k$. Also, let $i=\alpha-\ell=k-\beta$ and $j=\beta-\ell=k-\alpha$. Observe that $0\leq i,j\leq k-1$.

Suppose that $u_{-\ell}u_{-\ell+1}\dots u_{\ell}$ is a path such that $(u_{-\ell},u_{\ell})$ is $(i,j)$-poor. By the definition of $(i,j)$-poorness, the number of pairs of paths $(u_{-\ell}u_{-\ell-1}\dots u_{-\alpha},u_{\ell}u_{\ell+1}\dots u_{\beta})$ such that there exist $(|V(H)|+2)(2k+1)+1$ pairwise internally vertex-disjoint paths of length $\alpha+\beta=2k-i-j$ between $u_{-\alpha}$ and $u_{\beta}$ is $O(\delta^{i+j-1})$. Thus, the number of ways to extend $u_{-\ell}u_{-\ell+1}\dots u_{\ell}$ to a path $u_{-k}u_{-k+1}\dots u_{k}$ possessing property (ii) with our fixed choice of $\alpha$ and $\beta$ is $O(\delta^{i+j-1}\cdot(K\delta)^{k-\alpha+k-\beta})=O(\delta^{2k-2\ell-1})$, where the first factor bounds the number of possible ways to extend to $u_{-\alpha}u_{-\alpha+1}\dots u_{\beta}$, and the second factor bounds the number of possible ways to extend that to $u_{-k}u_{-k+1}\dots u_{k}$. The number of possible choices for $u_{-\ell}u_{-\ell+1}\dots u_{\ell}$ is $O(n\delta^{2\ell})$, so the result follows. ∎

We are now in a position to complete the proof of Lemma 2.6.

Proof of Lemma 2.6.

The condition $|S|=\omega(\frac{n}{\delta^{1/2}})$ implies that $n\delta^{2k-1}=o(\frac{|S|^{2}\delta^{2k}}{n})$, so by Lemmas 4.5 and 4.6, there are $\Omega(\frac{|S|^{2}\delta^{2k}}{n})$ paths $u_{-k}u_{-k+1}\dots u_{k}$ with both endpoints in $S$ such that $(u_{-\ell},u_{\ell})$ is $(i,j)$-poor for every $1\leq\ell\leq k$ and every $0\leq i,j\leq k-1$. Discard all those paths among these in which there is a subpath of length $k$ which is not $L$-good. By Lemma 4.7, we discarded $O(\frac{n\delta^{2k}}{L})$ paths, which is $o(\frac{|S|^{2}\delta^{2k}}{n})$, by the condition $|S|=\omega(\frac{n}{L^{1/2}})$. Of the remaining paths, discard all those for which there exist $1\leq\alpha,\beta\leq k$ with $\alpha+\beta>k$ such that there exist $(|V(H)|+2)(2k+1)+1$ pairwise internally vertex-disjoint paths of length $\alpha+\beta$ between $u_{-\alpha}$ and $u_{\beta}$. By Lemma 4.9, there are $O(n\delta^{2k-1})$ such paths, which is again $o(\frac{|S|^{2}\delta^{2k}}{n})$. Hence, we are left with $\Omega(\frac{|S|^{2}\delta^{2k}}{n})$ paths.

We claim that each such path is $L$-good. Suppose otherwise, and take a path $u_{-k}u_{-k+1}\dots u_{k}$ which is not $L$-good. Since each of its subpaths of length $k$ is $L$-good, by Lemma 4.8 there exist $1\leq\alpha,\beta\leq k$ with $\alpha+\beta>k$ such that there exist $(|V(H)|+2)(2k+1)+1$ pairwise internally vertex-disjoint paths of length $\alpha+\beta$ between $u_{-\alpha}$ and $u_{\beta}$. But we discarded these paths, which is a contradiction, and the proof is complete. ∎