Proof. By a well-known result of Erdős and Kleitman [Reference Erdős and Kleitman9], $\mathcal {F}$ contains a k-partite subgraph $\mathcal {F}_0$ with $|\mathcal {F}_0|\geq \frac {k!}{k^k}|\mathcal {F}|$ . Let $(X_1, X_2, \dots X_k)$ be a fixed k-partition of $\mathcal {F}_0$ . If $\mathcal {F}_0$ satisfies conditions 2 and 3, then the theorem holds with $\mathcal {F}'=\mathcal {F}_0$ . Otherwise, let $\mathcal {G}_0=\mathcal {F}_0$ . We perform a so-called filtering process on $\mathcal {G}_0$ as follows. Let $\mathcal {W}_0=\emptyset $ . For each $J\in \mathrm {Int}(\mathcal {G}_0)$ , let $\mathcal {A}_J=\emptyset $ . We iteratively modify $\mathcal {G}_0$ , $\mathcal {W}_0$ and the $\mathcal {A}_J$ ’s for $J\in \mathrm {Int}(\mathcal {G}_0)$ as follows. Whenever there is an edge $F\in \mathcal {G}_0$ and a $J\in \mathrm {Int}(\mathcal {G}_0)$ such that $\mathcal {L}_{\mathcal {G}_0}(F_J)$ is nonempty and not s-diverse, we remove all the edges of $\mathcal {G}_0$ containing $F_J$ and add them to $\mathcal {A}_J$ . Whenever there is an edge $F\in \mathcal {G}_0$ and a $J\in \mathrm {Int}(G_0)$ such that $\mathcal {L}_{\mathcal {G}_0}(F_j)$ is s-diverse but $d_{\mathcal {G}_0}(F_J)<\frac {1}{2k}\frac {|\mathcal {F}_0|}{n^{|J|}}$ , we remove all the edges of $\mathcal {G}_0$ containing $F_J$ and add them to $\mathcal {W}_0$ . Let $\mathcal {G}^*_0$ denote the final $\mathcal {G}_0$ at the end of the filtering process. By definition, if $\mathcal {G}^*_0$ is nonempty, then $\mathcal {G}^*_0$ satisfies conditions 2 and 3.

Note that $|\mathcal {W}_0|\leq |\mathcal {F}_0|/2$ . This is because for any fixed $j=1,\dots , k-1$ , there are at most $\binom {n}{j}$ different $F_J$ ’s. When all edges containing some $F_J$ are moved to $\mathcal {W}_0$ , by definition, fewer than $\frac {1}{2k}\frac {|\mathcal {F}_0|}{n^{|J|}}$ edges are moved. Hence $|\mathcal {F}_0\setminus \mathcal {W}_0|\geq |\mathcal {F}_0|/2$ . If $|\mathcal {G}_0^*|\geq \frac {1}{1+2^k} |\mathcal {F}_0\setminus \mathcal {W}_0|$ , we let $\mathcal {F}'=\mathcal {G}^*_0$ . Otherwise, by the pigeonhole principle, there exists some $J\in \mathrm {Int}(\mathcal {G}_0)$ such that $|\mathcal {A}_J|\geq \frac {1}{1+2^k} |\mathcal {F}_0\setminus \mathcal {W}_0|$ . Note that edges were added to $\mathcal {A}_J$ in batches, with each batch consisting of edges F with the same J-projection and different batches have different J-projections. Consider any batch $\mathcal {B}$ added to $\mathcal {A}_J$ . Let D denote the common J-projection of the edges in $\mathcal {B}$ . By definition, $\mathcal {L}_{\mathcal {B}}(D)=\mathcal {L}_{\mathcal {G}_0}(F_J)$ is nonempty and not s-diverse at the moment $\mathcal {B}$ was added to $\mathcal {A}_J$ . By definition, there exists a vertex v in $\mathcal {L}_{\mathcal {B}}(D)$ that lies in at least $(1/s)|\mathcal {L}_{\mathcal {B}}(D)|$ of the edges. Let $\mathcal {B}'$ denote the subset of edges in $\mathcal {B}$ that also contain v. We now remove $\mathcal {B}$ from $\mathcal {A}_J$ and replace it with $\mathcal {B}'$ . We do this for each batch of edges that were added to $\mathcal {A}_J$ , and denote the resulting subgraph of $\mathcal {A}_J$ by $\mathcal {A}^{\prime }_J$ . Then $|\mathcal {A}^{\prime }_J|\geq (1/s)|\mathcal {A}_J|$ . Furthermore, it is easy to see that $J\notin \mathrm {Int}(\mathcal {A}^{\prime }_J)$ . We let $\mathcal {F}_1=\mathcal {A}^{\prime }_J$ . Then

$$\begin{align*}|\mathcal{F}_1|\geq \frac{1}{s(1+2^k)}|\mathcal{F}_0\setminus \mathcal{W}_0|\geq \frac{1}{2s(1+2^k)}|\mathcal{F}_0| \quad \text{ and } \quad |\mathrm{Int}(\mathcal{F}_1)|\leq |\mathrm{Int}(\mathcal{F}_0)|-1.\end{align*}$$

Now, let $\mathcal {G}_1=\mathcal {F}_1$ , let $\mathcal {W}_1=\emptyset $ and set $\mathcal {A}_J=\emptyset $ for all $J\in \mathrm {Int}(\mathcal {G}_1)$ . We then perform the same filtering on $\mathcal {G}_1$ to iteratively modify $\mathcal {G}_1$ , $\mathcal {W}_1$ and the $\mathcal {A}_J$ ’s for $J\in \mathrm {Int}(\mathcal {G}_1)$ . Let $\mathcal {G}^*_1$ denote the final $\mathcal {G}_1$ at the end of the filtering process. By definition, $\mathcal {G}^*_1$ satisfies conditions 2 and 3. If $|\mathcal {G}_1^*|\geq \frac {1}{1+2^k}|\mathcal {F}_1\setminus \mathcal {W}_1|$ , we let $\mathcal {F}'=\mathcal {G}_1^*$ . Otherwise, as before, there exists some $J\in \mathrm {Int}(\mathcal {G}_1)$ and a subgraph $\mathcal {A}^{\prime }_J\subseteq \mathcal {A}_J$ with $|\mathcal {A}^{\prime }_J|\geq \frac {1}{2s(1+2^k)} |\mathcal {F}_1|$ such that $|\mathrm {Int}(\mathcal {A}^{\prime }_J)|\leq |\mathrm {Int}(\mathcal {G}_1)|-1$ . We let $\mathcal {F}_2=\mathcal {A}^{\prime }_J$ .

We continue like this, obtaining a sequence $\mathcal {F}_0, \mathcal {F}_1,\dots $ . Since $|\mathrm {Int}(\mathcal {F}_i)|$ strictly decreases with i, the sequence must end with $\mathcal {F}_m$ for some $m\leq 2^k-1$ . Since $\mathcal {F}_{m+1}$ is undefined, this must mean that

$$\begin{align*}|\mathcal{G}_m^*|\geq \frac{1}{1+2^k}|\mathcal{F}_m\setminus \mathcal{W}_m|\geq \frac{1}{2(1+2^k)}|\mathcal{F}_m|\geq \frac{1}{[2s(1+2^k)]^{2^k}}|\mathcal{F}_0|\geq c(k,s)|\mathcal{F}|,\end{align*}$$

where $c(k,s)=\frac {k!}{k^k}\cdot \frac {1}{[2s(1+2^k)]^{2^k}}$ . Let $\mathcal {F}'=\mathcal {G}^*_m$ . Then $|\mathcal {F}'|\geq c(k,s)|\mathcal {F}|$ and $\mathcal {F}'$ also satisfies conditions 2 and 3, by the definition of $\mathcal {G}^*_m$ .

Proof. Statement 1 follows from definition of $\mathrm {Int}(\mathcal {F})$ . For statement 2, let $J,J'\in \mathrm {Int}(\mathcal {F})$ with $J\neq J'$ . Let $F\in \mathcal {F}$ . By our assumption $\mathcal {L}_{\mathcal {F}}(F_J)$ is s-diverse. So the vertices in $F\setminus F_J$ block fewer than $k(1/s)|\mathcal {L}_{\mathcal {F}}(F_J)| \leq |\mathcal {L}_{\mathcal {F}}(F_J)|$ of the edges in $\mathcal {L}_{\mathcal {F}}(F_J)$ . So there exists $F'\in \mathcal {F}$ with $F\cap F'=F_J$ . By a similar reasoning, since $2k(1/s)|\mathcal {L}_{\mathcal {F}}(F^{\prime }_{J'})| \leq |\mathcal {L}_{\mathcal {F}}(F^{\prime }_{J'})|$ there exists $F"\in \mathcal {F}$ containing $F^{\prime }_{J'}$ that avoids vertices in $(F\setminus F')\cup (F'\setminus F^{\prime }_{J'})$ . Now, $\pi (F\cap F")=J\cap J'$ . Hence, $J\cap J'\in \mathrm {Int}(\mathcal {F})$ .

For statement 3, suppose $\mathrm {Int}(\mathcal {F})$ has rank m. Then $\exists D\subseteq [k]$ with $|D|=m$ such that D is not contained in any member of $\mathrm {Int}(\mathcal {F})$ . Consider any $F,F'\in \mathcal {F}$ , where $F\neq G$ . If $F[D]=F'[D]$ , then $\pi (F\cap F')$ is a member of $\mathrm {Int}(\mathcal {F})$ that contains D, a contradiction. So, the D-projections of members of $\mathcal {F}$ are all distinct. This implies that $|\mathcal {F}|\leq \binom {n}{m}$ .

Proof. If $r(\mathcal {J})=k$ then $[k]\setminus \{j\}\in \mathcal {J}$ for each $j\in [k]$ . Since $\mathcal {J}$ is closed under intersection, we see that $S\in \mathcal {J}$ for each proper subset S of $[k]$ . Hence statement 1 clearly holds. Next, suppose $r(\mathcal {J})=k-1$ . Then some $(k-1)$ -subset of $[k]$ is not in $\mathcal {J}$ . Without loss of generality, suppose $[k]\setminus \{j\}\notin \mathcal {J}$ for $j=1,\dots , t$ and $[k]\setminus \{j\}\in \mathcal {J}$ for $j=t+1,\dots , k$ , for some $1\leq t\leq k$ .

First, suppose $t=1$ . Then $[k]\setminus \{1\}\notin \mathcal {J}$ and $[k]\setminus \{2\},\dots , [k]\setminus \{k\}\in \mathcal {J}$ . Since $\mathcal {J}$ is closed under intersection, every proper subset of $[k]$ that contains $1$ is in $\mathcal {J}$ . We already have $[k]\setminus \{1\}\notin \mathcal {J}$ . Suppose there is a $(k-2)$ -subset B of $[k]\setminus \{1\}$ that is in $\mathcal {J}$ . Let S be any subset of B. By earlier discussion $S\cup \{1\}\in \mathcal {J}$ . Since $\mathcal {J}$ is closed under intersection, we have $S=(S\cup \{1\})\cap B\in \mathcal {J}$ . Hence, $2^B\subseteq \mathcal {J}$ . So statement 1 holds in this case. Hence, we may assume that no $(k-2)$ -subset of $[k]\setminus \{1\}$ is in $\mathcal {J}$ . Then statement 2 holds for $i=1$ . It is easy to see that if an i satisfies the requirements, it can only be $1$ .

Next, suppose $t\geq 2$ . Observe that for any $1\leq i<j\leq t$ , we must have $[k]\setminus \{i,j\}\in \mathcal {J}$ , as otherwise $[k]\setminus \{i,j\}$ is not contained in any member of $\mathcal {J}$ , contradicting $r(\mathcal {J})\geq k-1$ . Also, by our assumption for every $t+1\leq j\leq k$ we have $[k]\setminus \{j\}\in \mathcal {J}$ . Since $\mathcal {J}$ is closed under intersection, we see that every subset of $[k]\setminus \{1,2\}$ is in $\mathcal {J}$ . So statement 1 holds.

Proof. First we apply Lemma 3.1 to $\mathcal {F}$ to get an s-super-homogeneous subgraph $\mathcal {F}_1$ of size at least $c(k,s)|\mathcal {F}|$ . If $|\mathcal {F}\setminus \mathcal {F}_1|\leq \frac {1}{2} \varepsilon |\mathcal {F}|$ , then we stop. Otherwise, we apply the lemma to $\mathcal {F}\setminus \mathcal {F}_1$ to find an s-super-homogeneous subgraph $\mathcal {F}_2$ of size at least $c(k,s)|\mathcal {F}\setminus \mathcal {F}_1|$ . In general, as long as $|\mathcal {F}\setminus (\mathcal {F}_1\cup \cdots \cup \mathcal {F}_i)|>\frac {1}{2}\varepsilon |\mathcal {F}|$ , we let $\mathcal {F}_{i+1}$ be an s-super-homogeneous subgraph of $\mathcal {F}\setminus (\mathcal {F}_1\cup \cdots \cup \mathcal {F}_i)$ of size at least $c(k,s)|\mathcal {F}\setminus (\mathcal {F}_1\cup \cdots \cup \mathcal {F}_i)|$ , which exists by Lemma 3.1. Suppose the sequence of subgraphs we constructed are $\mathcal {F}_1,\dots , \mathcal {F}_m$ .

By our algorithm, for each $i\in [m]$ , $|\mathcal {F}_i|\geq \frac {1}{2}c(k,s) \varepsilon |\mathcal {F}|$ . Note that this implies that $m\leq \frac {2}{c(k,s)\varepsilon }$ . For each $i\in [m]$ , by Lemma 3.3 part 3, $r(\mathrm {Int}(\mathcal {F}_i))\geq k-1$ and by Lemma 3.3 part 2 and Lemma 3.4, $\mathrm {Int}(\mathcal {F}_i)$ is either of type 1 or is of type 2. If for some $i\in [m]$ , $\mathrm {Int}(\mathcal {F}_i)$ is of type 1, then we let $\mathcal {F}'=\mathcal {F}_i$ for such an i and Statement 1 holds. Hence, we may assume that for each $i\in [m]$ , $\mathrm {Int}(\mathcal {F}_i)$ is of type 2. Let $\mathcal {F}^*=\bigcup _{j=1}^m \mathcal {F}_j$ . Then

$$\begin{align*}|\mathcal{F}^*|\geq (1-\frac{\varepsilon}{2})|\mathcal{F}|.\end{align*}$$

Set $h=\left \lfloor \left (\frac {5m}{\varepsilon }\right )^{k-1} \right \rfloor $ . Since $m\leq \frac {2}{c(k,s)\varepsilon }$ , we have

(7) $$ \begin{align} h\leq \left(\frac{10}{c(k,s)\varepsilon^2}\right)^{k-1}. \end{align} $$

For each $j\in [m]$ , let $i_j\in [k]$ denote the central index for $\mathrm {Int}(\mathcal {F}_j)$ and for each $F\in \mathcal {F}_j$ , let $c(F) = F_{\{i_j\}}$ . We call $c(F)$ the central vertex of F. We partition $\mathcal {F}^*$ according to $c(F)$ . For each $i \in [n]$ , let

$$\begin{align*}\mathcal{A}_i = \{ F \in \mathcal{F}^* : c(F) = i\} \text{ and } \mathcal{A}_i' = \{F \setminus \{i\}: F \in \mathcal{A}_i\}.\end{align*}$$

Clearly, for each $i\in [n]$ we have $|\mathcal {A}_i|=|\mathcal {A}^{\prime }_i|$ . For each $j \in [m]$ , let

$$\begin{align*}\mathcal{F}_j' = \{ F \setminus \{c(F)\} : F \in \mathcal{F}_j\}=\{F_{[k]\setminus \{i_j\}}: F\in \mathcal{F}_j\}.\end{align*}$$

Then, in particular:

$$\begin{align*}\bigcup_{i = 1}^n \mathcal{A}_i' = \bigcup_{j = 1}^m \mathcal{F}_j'. \end{align*}$$

The next claim is crucial to our argument.

Claim 3.6. For each $j\in [m]$ , we have

$$\begin{align*}|\partial_{k-2}(\mathcal{F}_j')|=\sum_{i = 1}^n |\partial_{k - 2} (\mathcal{A}_i' \cap \mathcal{F}_j')|.\end{align*}$$

Note that trivially $|\partial _{k - 2} (\mathcal {F}_j')| \leq \binom {n}{k - 2}$ . Now, we have

(8) $$ \begin{align} \sum_{i = 1}^n |\partial_{k - 2}(\mathcal{A}_i')| \leq \sum_{i = 1}^n\sum_{j = 1}^m |\partial_{k - 2}(\mathcal{A}_i' \cap \mathcal{F}_j')|= \sum_{j = 1}^m \sum_{i = 1}^n|\partial_{k - 2}(\mathcal{A}_i' \cap \mathcal{F}_j')| = \sum_{j =1}^m |\partial_{k - 2} (\mathcal{F}_j')|\leq m \binom{n}{k - 2} \end{align} $$

For each $i \in [n]$ , let $x_i$ be the real such that $|\partial _{k - 2}(\mathcal {A}_i')| = \binom {x_i}{k - 2}$ , where without loss of generality, $x_1 \geq x_2 \dots \geq x_n$ . By (6), for each $i\in [n]$ we have $|\mathcal {A}_i'| \leq \binom {x_i}{k - 1}$ .

By (8), $\sum _{i=1}^n\binom {x_i}{k-2}\leq m\binom {n}{k-2}$ . Thus, $\binom {x_{h+1}}{k - 2} \leq \frac {m}{h+1} \binom {n}{ k - 2}$ , which yields

(9) $$ \begin{align} x_{h+1} - k + 3 \leq \left( \frac{m}{h+1} \right)^{ 1 / (k - 2)} n. \end{align} $$

Thus, we have

$$ \begin{align*} \sum_{i> h} |\mathcal{A}_i| =\sum_{i>h}|\mathcal{A}^{\prime}_i|&\leq \sum_{i = h + 1}^n \binom{x_i}{k - 1} \\&=\sum_{h+1}^n \frac{x_i-k+2}{k-1}\binom{x_i}{k-2}\\&\leq \frac{x_{h+1} - k +2 }{k - 1} \sum_{i = h + 1}^{n} \binom{x_i}{k - 2} \\&\leq \left( \frac{m}{h+1 } \right)^{ 1 / (k - 2)} \frac{nm}{k - 1} \binom{n}{k - 2} \quad \quad \text{(by (}{8}\text{) and (}{9}))\\\end{align*} $$

By our choice of h, $h+1\geq (\frac {5m}{\varepsilon })^{k-1}$ . So we have $\sum _{i>h} |\mathcal {A}_i|<\left (\frac {\varepsilon }{5}\right )^{1 + \frac {1}{k - 2}}\frac {n}{k-1}\binom {n}{k-2}\leq \frac {\varepsilon }{4} a\binom {n}{k-1}$ , for sufficiently large n and by our assumption $a \geq \varepsilon ^{\frac {1}{ k - 2}}$ . Let $W := [h]$ . Let $\mathcal {G}_1:= \{ F \in \mathcal {F}^*: c(F) \not \in W\}$ and $\mathcal {G}_2:= \{F \in \mathcal {F}^*: |F \cap W| \geq 2\}$ .

Then by the argument above, $|\mathcal {G}_1|=\sum _{i>h}|\mathcal {A}_i| \leq \frac {\varepsilon }{4} a \binom {n}{k - 1}$ . Also, for n sufficiently large, we have $|\mathcal {G}_2| \leq \binom {W}{2} \binom {n}{ k - 2} \leq \frac {\varepsilon }{4} a\binom {n}{k-1}$ . Let $\mathcal {F}":= \mathcal {F}^* \setminus (\mathcal {G}_1 \cup \mathcal {G}_2)$ . We can readily check that $\mathcal {F}"$ satisfy all of conditions 1,2,3.

Proof. We iteratively remove edges of $\mathcal {F}$ to form $\mathcal {F}'$ in accordance with the following process. Whenever there is some $B \subseteq V(\mathcal {F})$ with $|B|\leq k-1$ such that B is contained in at least one but fewer than $\frac {1}{2 k } \frac {|\mathcal {F}|}{\binom {n}{|B|}}$ edges, we mark B and remove all edges containing B from $\mathcal {F}'$ . Clearly at some point this process will terminate. We denote the final graph by $\mathcal {F}'$ .

Let us count now how many edges $\mathcal {F}'$ has. For each $i=1,\dots , k-1$ , there are at most $\binom {n}{i}$ marked i-sets. For each marked i-set, we have removed at most $\frac {1}{2k}\frac {|\mathcal {F}|}{\binom {n}{i}}$ edges from $\mathcal {F}$ . Summing over $i=1, \dots , k-1$ , we see that we have removed at most $\frac {1}{2} |\mathcal {F}|$ edges. Hence, $|\mathcal {F}'| \geq \frac {1}{2} |\mathcal {F}|$ . Since the process terminates with a nonempty $\mathcal {F}'$ , it must be the case that for each $i=1,\dots , k-1$ and any i-set contained in an edge of $\mathcal {F}'$ it is in at least $\frac {1}{2k}\frac {|\mathcal {F}|}{\binom {n}{i}}$ edges, completing the proof.

Proof. By picking $\eta $ sufficiently small, we may assume n is sufficiently large in terms of s and k. By Lemma 4.1, $\mathcal {F}$ contains a subgraph $\mathcal {F}'$ such that $|\mathcal {F}'|\geq \frac {1}{2}|\mathcal {F}|$ and for each $i\in [k-1]$ , $\delta _i(\mathcal {F}')\geq \frac {1}{2k} \frac {|\mathcal {F}|}{\binom {n}{i}}=\frac {a}{2k}\frac {\binom {n}{k-1}}{\binom {n}{i}}$ . Let $E_1,\dots , E_t$ be a tree-defining ordering of $\mathcal {H}$ . For each $j\in [t]$ , let $\mathcal {H}_j=\{E_1,\dots , E_j\}$ . We prove by induction that for each $j=1,\dots , t$ , $\mathcal {F}$ contains at least $(\frac {a}{8k!})^jn^{v(\mathcal {H}_j)-j}$ copies of $\mathcal {H}_j$ .

For the base case, $\mathcal {F}'$ contains at least $\frac {1}{2}|\mathcal {F}|=\frac {a}{2}\binom {n}{k-1}\geq \frac {a}{8k!} n^{v(\mathcal {H}_1)-1}$ copies of $\mathcal {H}_1$ . So the claim holds. For the induction step, let $1\leq j\leq t-1$ and suppose $\mathcal {F}'$ contains at least $(\frac {a}{8k!})^j n^{v(\mathcal {H}_j)-j} $ copies of $\mathcal {H}_j$ . Let F denote a parent edge of $E_{j+1}$ in $\mathcal {H}_j$ . Let $L=F\cap E_{j+1}=V(\mathcal {H}_j)\cap E_{j+1}$ and $\ell =|L|$ . Let $\mathcal {H}'$ denote a copy of $\mathcal {H}_j$ in $\mathcal {F}'$ and $L'$ the image of L in $\mathcal {H}'$ . By our assumptions of $\mathcal {F}'$ , $\deg _{\mathcal {F}'}(L') \geq \frac {a}{2k} \frac {\binom {n}{k - 1}}{\binom {n}{\ell }}$ . For each vertex $v \in V(\mathcal {H}')$ , we have that v is contained in at most $\binom {n - \ell - 1}{k - \ell - 1}$ edges which contain $L'$ . As there are at most s vertices in $\mathcal {H}_j$ , we have that for sufficiently large n the number of edges of $\mathcal {F}'$ that contain $L'$ but do not intersect $\mathcal {H}'$ somewhere else is at least

$$ \begin{align*} \frac{a}{2k} \frac{\binom{n}{k - 1}}{\binom{n}{\ell}} - s \binom{n - \ell - 1}{k - \ell - 1} \geq \frac{a \ell !}{4 k!} n^{k - 1 - \ell} - \frac{s}{(k - \ell - 1)!} n^{k - \ell - 1} \geq \frac{a}{8k!}n^{k - 1 - \ell}, \end{align*} $$

where we used the fact that $a\geq 8sk!$ . Hence $\mathcal {H}'$ can be extended to at least $\frac {a}{8k!} n^{k-1-\ell }$ copies of $\mathcal {H}_{j+1}$ in $\mathcal {F}'$ . Therefore, $\mathcal {F}'$ contains at least $(\frac {a}{8k!})^{j+1}n^{v(\mathcal {H}_j)-j+k-1-\ell }=\frac {a}{8k!}^{j+1}n^{v(\mathcal {H}_{j+1})- (j+1)}$ copies of $\mathcal {H}_{j+1}$ . This completes the induction and the proof.

Proof. Let $(X_1,\dots , X_k)$ be a k-partition associated with $\mathrm {Int}(\mathcal {F})$ . Without loss of generality, suppose $2^{[k-2]}\subseteq \mathrm {Int}(\mathcal {F})$ . Since $\mathcal {H}$ is $2$ -contractible, each edge E contains at least two vertices of degree $1$ . We will designate two of them as expansion vertices for E.

Let $E_1,\cdots , E_t$ be a tree-defining ordering of $\mathcal {H}$ . For each $i\in [t]$ , let $\mathcal {H}_i=\{E_1,\dots , E_i\}$ , $v_i=|V(\mathcal {H}_i)|$ , and let $\Pi _i$ denote the set of injective homomorphisms $\varphi ^i$ of $\mathcal {H}_i$ into $\mathcal {F}$ that maps all the expansion vertices in $\mathcal {H}_i$ into $X_{k-1}\cup X_k$ . We will prove by induction the stronger statement that for each $i\in [t]$ , $|\Pi _i|\geq (\frac {\varepsilon }{2ks})^i n^{v_i-i}$ . The base step $i=1$ is trivial since $\mathcal {F}$ has at least $\varepsilon n^{k-1}$ edges E and for each E we can define an embedding $\varphi ^1$ of $\mathcal {H}_1$ to E where the expansion vertices are mapped to $X_{k-1}\cup X_k$ . For the induction step, let $1\leq i\leq t-1$ and suppose $|\Pi _i|\geq (\frac {\varepsilon }{2ks})^i n^{v_i-i}$ . Let F be a parent edge of $E_{i+1}$ in $\mathcal {H}_i$ . Consider any $\varphi ^i\in \Pi _i$ . Since vertices in $F\cap E_{i+1}$ have degree at least two in $\mathcal {H}$ and are not expansion vertices in $\mathcal {H}_i$ , we have $\varphi ^i(F\cap E_{i+1})\subseteq \bigcup _{j=1}^{k-2} X_j$ . Since $2^{[k-2]}\subseteq \mathrm {Int}(\mathcal {F})$ , we have $\pi (\varphi ^i(F\cap E_{i+1})) \in \mathrm {Int}(\mathcal {F})$ . So, $\mathcal {L}_{\mathcal {F}}(\varphi ^i(F \cap E_{i+1}))$ is s-diverse. Therefore, there are at least $\frac {s -v_i}{s} | \mathcal {L}_{\mathcal {F}}( \varphi ^i (F\cap E_{i+1}))|\geq \frac {1}{s} | \mathcal {L}_{\mathcal {F}}( \varphi ^i (F\cap E_{i+1}))|$ edges of $\mathcal {F}$ that contain $\varphi ^i(F\cap E_{i+1})$ and do not intersect the rest of $\varphi ^i(\mathcal {H}_i)$ . Let $\ell =|F\cap E_{i+1}|$ . Since $\mathcal {F}$ is s-super-homogeneous with $\mathrm {Int}(\mathcal {F})\supseteq 2^{[k-2]}$ and $\pi (\varphi ^i(F\cap E_{i+1}))\in \mathrm {Int}(\mathcal {F})$ , by the conditions of s-super-homogeneous given in Theorem 3.1, $|\mathcal {L}(\varphi ^i (F \cap E_{i+1}))| \geq \frac {|\mathcal {F}|}{2kn^\ell } \geq \frac {\varepsilon }{2k} n^{ k - \ell -1}$ . Hence each mapping $\varphi _i$ in $\Pi _i$ can be extended to at least $\frac {\varepsilon }{2ks} n^{ k -\ell - 1}$ injective mappings $\varphi ^{i+1}$ of $\mathcal {H}_{i+1}$ , by mapping $E_{i+1}$ to an edge $F'$ of $\mathcal {F}$ that contains $\varphi ^i(F\cap E_{i+1})$ and avoids the rest of $\varphi ^i(\mathcal {H}_i)$ . Furthermore, we can require $\varphi ^{i+1}$ to map the two designated expansion vertices of $E_{i+1}$ to $X_{k-1}\cup X_k$ . Hence, $|\Pi _{i+1}|\geq |\Pi _i|\cdot \frac {\varepsilon }{2ks} n^{ k -\ell - 1} \geq (\frac {\varepsilon }{2ks})^{i+1} n^{v_{i+1}-(i+1)}$ . This completes the induction step and the proof.

Proof. For convenience, let $\sigma =\sigma (\mathcal {H})$ . Note that if $a \geq 8 s k!$ , we may apply Lemma 4.2, so at the cost of a constant factor on $\beta $ , we will assume $a = \sigma - 1 + \gamma $ . We apply Theorem 3.5 to $\mathcal {F}$ with $\varepsilon = \frac {\gamma }{4 \sigma }$ if $\sigma \geq 2$ and $\varepsilon = \frac {1}{4} \gamma ^{k - 2}$ if $\sigma = 1$ . Then one of the following two cases holds.

Case 1. $\mathcal {F}$ contains an s-super-homogeneous subgraph $\mathcal {F}'$ such that $|\mathcal {F}'|\geq \frac {1}{2} c(k,s)\varepsilon |\mathcal {F}|$ and $\mathrm {Int}(\mathcal {F}')$ is of type 1.

By Lemma 4.3, $\mathcal {F}'$ contains $\left ( \frac {\varepsilon c(k, s) }{2 ks}\right )^t n^{s - t}$ copies of $\mathcal {H}$ . So the claim holds.

Case 2. There exists a $W \subseteq [n]$ and $\mathcal {F}" \subseteq \mathcal {F}$ satisfying the following:

1. (a)

   $$ \begin{align*}|W| \leq \left( \frac{160 \sigma^2}{c(k,s) \gamma^{2k - 4}} \right)^{k - 1}.\end{align*} $$

2. (b)

   $$ \begin{align*}|\mathcal{F}"| \geq \left(1-\frac{\gamma}{4\sigma}\right)|\mathcal{F}|.\end{align*} $$

3. (c) Every $F \in \mathcal {F}"$ satisfies $|W \cap F| = 1$ .

Observe that in $\sigma = 1$ case, Theorem 3.5 gives the upper bound $W \leq \left ( \frac {160}{c(k, s) \gamma ^{ 2k - 4} } \right )^{ k -1}$ and in $\sigma \geq 2$ case, gives the bound $W \leq \left ( \frac {160 \sigma ^2}{c(k, s) \gamma ^{ 2} } \right )^{ k -1} $ . In both cases, the written upper bound holds. For simplicity, observe that for both choices of $\varepsilon $ , $|\mathcal {F}"| \geq (1 - \varepsilon )|\mathcal {F}| \geq \left ( 1 - \frac {\gamma }{4 \sigma }\right ) |\mathcal {F}|$ .

For each $S\in \binom {W}{\sigma }$ , let $\mathcal {L}^*(S) = \bigcap _{v \in S} \mathcal {L}_{\mathcal {F}"}(v)$ . We say a set $S \in \binom {W}{\sigma }$ is good, if $|\mathcal {L}^*(S)| \geq 8s (k - 1)! \binom {n}{k-2}$ ; otherwise we say S is bad. Let $\mathcal {D}$ denote the set of $(k-1)$ -sets D in $[n]\setminus W$ with $d_{\mathcal {F}"}(D)\geq \sigma $ . Since each edge of $\mathcal {F}"$ contains one vertex in W and a $(k-1)$ -set in $[n]\setminus W$ , via double counting, we get

(10) $$ \begin{align} \sum_{S \in \binom{W}{\sigma}} |\mathcal{L}^*(S)| = \sum_{D \in \mathcal{D}} \binom{d_{\mathcal{F}"}(D)}{\sigma}. \end{align} $$

By our assumption, $|\mathcal {F}"|\geq (1-\frac {\gamma }{4 \sigma })|\mathcal {F}|\geq (1-\frac {\gamma }{4\sigma })(\sigma -1+\gamma )\binom {n}{k-1}\geq (\sigma -1+\frac {\gamma }{2})\binom {n}{k-1}$ . This implies that $m:=\sum _{D\in \mathcal {D}} d_{\mathcal {F}"} (D)\geq \frac {\gamma }{2}\binom {n}{k-1}$ . Observe that for all $D \in \mathcal {D}$ , $\binom {d_{\mathcal {F}"}(D)}{\sigma } \geq \frac {d_{\mathcal {F}"}(D)}{\sigma } $ .

Thus,

$$ \begin{align*}\sum_{S \in \binom{W}{\sigma}} |\mathcal{L}^*(S)| \geq \frac{m}{\sigma} \geq \frac{\gamma}{2 \sigma }\binom{n}{ k - 1}.\end{align*} $$

On the other hand, the contribution to the left-hand side of (10) from the bad $\sigma $ -sets of W is at most $\binom {|W|}{\sigma } 8s(k-1)!\binom {n}{k-2}<\frac {\gamma }{4\sigma }\binom {n}{k-1}$ , for sufficiently large n. Hence, we have

$$\begin{align*}\sum_{S\in \binom{W}{\sigma}, \, S \text{ good}} |\mathcal{L}^*(S)| \geq \frac{\gamma}{4\sigma} \binom{n}{ k - 1}.\end{align*}$$

Fix any $S\in \binom {|W|}{\sigma }$ that is good, let $d(S) = \frac {|\mathcal {L}^*(S)|}{\binom {n}{ k - 2}}$ , in particular, since S is good, $d(S) \geq 8s (k - 1)!$ . Let R be a minimum cross-cut of $\mathcal {H}$ . Let $\mathcal {H}'=\mathcal {H}-R$ . It is easy to see that $\mathcal {H}'$ is a $(k-1)$ -tree. Observe that any copy of $\mathcal {H}'$ in $\mathcal {L}^*(S)$ corresponds to a copy of $\mathcal {H}$ in $\mathcal {F}$ by joining it to S and different copies of $\mathcal {H}'$ in $\mathcal {L}^*(S)$ give rise to different copies of $\mathcal {H}$ . Let $t'=|\mathcal {H}'|$ and note that $v(\mathcal {H}')=s-\sigma $ . By Lemma 4.2, for every good S, there are at least $\eta d(S)^{t'}n^{s - \sigma - t'}$ many copies of $\mathcal {H}'$ in $\mathcal {L}^*(S)$ . Thus, the number of copies of $\mathcal {H}$ in $\mathcal {F}$ is at least:

$$ \begin{align*} \sum_{S\in \binom{W}{\sigma}, \, S \text{ good }} \eta d(S)^{t'} n^{s - \sigma - t'} &\geq \eta n^{s - \sigma - t' - (k - 2)t'} \sum_{S\in \binom{W}{\sigma},\, S \text{ good }} |\mathcal{L}^*(S))|^{t'} \\ &\geq \eta n^{s - \sigma - t' - (k - 2)t'} |W|^{\sigma} \left( \frac{1}{|W|^{\sigma}} \sum_{S \text{ good }} |\mathcal{L}^*(S)|\right) ^{t'}\\ &\geq \eta n^{s - \sigma - t' - (k - 2)t'} |W|^{\sigma} \left( \frac{1}{|W|^{\sigma}} \frac{\gamma}{4 \sigma} \binom{n}{ k - 1}\right) ^{t'}\\ &\geq \beta a^t n^{s - \sigma}\geq \beta a^t n^{s-t}, \end{align*} $$

for some constant $\beta>0$ , where we used the fact that $a=\sigma -1+\gamma $ is a constant and $|W|$ is bounded by a constant.

Proof. Let $B = \{ v\in V(\mathcal {K}): d_{\mathcal {K}}(v) = \left \lfloor d^{t - 1} \right \rfloor \}$ . Then clearly,

$$\begin{align*}|B| \left\lfloor d^{t - 1} \right\rfloor \leq t|\mathcal{K}|.\end{align*}$$

Since $|\mathcal {K}|\leq d^{t-1}M$ , we have $|B| \leq 2 t M $ .

Now, consider any admissible set $S\subseteq V(\mathcal {K})$ with $|S|\leq t - 1$ . For every $D \subseteq S$ , let $\mathcal {B}(D) := \{ v \in V(\mathcal {K}) : D \cup \{ v\} \text { is saturated} \}$ . By definition, $\mathcal {Z}_{\mathcal {K}}(S) = \bigcup _{D \subseteq S, D \neq \emptyset } \mathcal {B}(D)$ . Consider a fixed nonempty $D\subseteq S$ . Note that

(11) $$ \begin{align} \sum_{v \in \mathcal{B}(D)}d_{\mathcal{K}}( D\cup \{v \}) \leq t d_{\mathcal{K}}(D), \end{align} $$

since each edge of $\mathcal {K}$ containing D is counted at most t times in the sum on the left-hand side. Since S is admissible, D is not saturated. Thus, $d_{\mathcal {K}}(D) \leq d^{t - |D|}$ . On the other hand, for every $v \in \mathcal {B}(D)$ , $D\cup \{v\}$ is saturated. So, $d_{\mathcal {K}}(D \cup \{v\}) = \left \lfloor d^{t - |D| - 1} \right \rfloor $ . By (11), we get $|\mathcal {B}(D)|\left \lfloor d^{t-|D|-1} \right \rfloor \leq t d^{t-|D|}$ , implying

$$ \begin{align*}|\mathcal{B}(D)| \leq 2 t d.\end{align*} $$

Summing over all $D \subseteq S$ with $D \neq \emptyset $ , we get $|\mathcal {Z}(S)|\leq t2^{t+1}d$ .

Proof. Let $\beta $ be sufficiently small and $a_0$ be sufficiently large. Let $\mathcal {K}$ be maximal collection of copies of $\mathcal {H}$ in $\mathcal {F}$ that satisfies condition 2. Since any collection that consists of a single copy of $\mathcal {H}$ satisfies condition 2, $\mathcal {K}$ exists. We show that $\mathcal {K}$ must also satisfy condition 1. Suppose on the contrary that $|\mathcal {K}| < \beta ^t n^{ k - 1} (a n^{k - \ell })^{t - 1}$ . To derive a contradiction, it suffices to show that $\mathcal {F}$ contains a copy $\mathcal {H}^*$ of $\mathcal {H}$ that is admissible relative to $\mathcal {K}$ . Indeed, if we found such an $\mathcal {H}^*$ , in particular, it would not be saturated by $\mathcal {K}$ . Hence, $d_{\mathcal {K}}(\mathcal {H}^*) < (\beta a n^{k-\ell -1})^{t-t}=1$ . So, $\mathcal {H}^*\notin \mathcal {K}$ . Furthermore, for all $S \subseteq \mathcal {H}^*$ , we would have $d_{\mathcal {K}}(S) \leq \left \lfloor (\beta a n^{k - \ell })^{t - |S|} \right \rfloor - 1$ . Thus, $d_{\mathcal {K}\cup \mathcal {H}^*}(S)\leq (\beta a n^{k - \ell })^{t - |S|}$ . Therefore, $\mathcal {K}\cup \{\mathcal {H}^*\}$ would still satisfy condition 2, which would contradict the maximality of $\mathcal {K}$ and complete the proof. Thus, it suffices to find such an admissible $\mathcal {H}^*$ .

Since $\mathcal {K}$ satisfies condition 2, $\mathcal {K}$ is $\beta a n^{ k- \ell - 1}$ -graded. Recall that a set $S \subseteq \mathcal {F}$ is saturated by $\mathcal {K}$ if $d_{\mathcal {K}}(S) = \left \lfloor (\beta an^{k - \ell })^{t - |S|} \right \rfloor $ . Let $\mathcal {F}_{\mathrm {heavy}} := \{F \in \mathcal {F}: d_{\mathcal {K}}(F) = \left \lfloor (\beta an^{k - \ell })^{t - 1} \right \rfloor \}$ , so that $\mathcal {F}_{\mathrm {heavy}}$ consists of all the saturated edges of $\mathcal {F}$ . By Lemma 5.1, $|\mathcal {F}_{\mathrm {heavy}}| \leq 2t \beta a n^{ k - 1}$ . Let $\mathcal {F}^* = \mathcal {F} \setminus \mathcal {F}_{\mathrm {heavy}}$ . Then $|\mathcal {F}^*|\geq \frac {1}{2}|\mathcal {F}|$ , as long as $\beta < \frac {1}{8 t (k - 1)!}$ . By definition, every edge in $\mathcal {F}^*$ is admissible relative to $\mathcal {K}$ . By Lemma 4.1, there exists $\mathcal {F}' \subseteq \mathcal {F}^*$ such that $|\mathcal {F}'| \geq \frac {1}{4} |\mathcal {F}|$ and for each $i\in [k-1]$ , $\delta _i(\mathcal {F}')\geq \frac {a}{4 k } \frac {\binom {n}{k - 1}}{\binom {n}{ i}}$ .

For each $i\in [t]$ , let $\mathcal {H}_i=\{E_1,\dots , E_i\}$ . We will inductively find embeddings $\varphi ^i$ of $\mathcal {H}_i$ into $\mathcal {F}'$ , such that $\varphi ^i(\mathcal {H}_i)$ is admissible relative to $\mathcal {K}$ . First, let $F_1$ be any edge of $\mathcal {F}'$ and let $\varphi ^1$ be any bijection from the vertices of $E_1$ to the vertices of $F_1$ . Since $F_1$ is admissible relative to $\mathcal {K}$ , $\varphi ^1(\mathcal {H}_1)$ is admissible. Let $1\leq i\leq t-1$ and suppose we have defined an embedding $\varphi ^i$ of $\mathcal {H}_i$ into $\mathcal {F}'$ such that $\varphi ^i(\mathcal {H}_i)$ admissible relative to $\mathcal {K}$ . Let F denote a parent edge of $E_{i+1}$ in $\mathcal {H}_i$ . Let $p=|F\cap E_{i+1}|$ . Since $\mathcal {H}$ is $\ell $ -overlapping, $p\leq \ell $ . By our conditions on $\mathcal {F}'$ , we have $d_{\mathcal {F}'}(\varphi ^i(F\cap E_{i + 1})) \geq \frac {a}{4k} \frac {\binom {n}{k - 1}}{\binom {n}{p}}$ . The number of edges of $\mathcal {F}'$ that contain $\varphi ^i(F\cap E_{i + 1})$ and some other vertex of $\varphi ^i(\mathcal {H}_i)$ is at most $kt \binom {n}{ k - p - 1}$ . Since $\mathcal {K}$ is $\beta a n^{ k- \ell - 1}$ -graded, by Lemma 5.1, we have $\mathcal {Z}_{\mathcal {K}}(\varphi ^i(\mathcal {H}_i)),$ the set of edges E of $\mathcal {F}'$ such that $\{E\} \cup \varphi ^i(\mathcal {H}_i)$ is inadmissible relative to $\mathcal {K}$ , has size at most $t 2^{t + 1} \beta a n^{ k -\ell - 1}$ .

Therefore as long as $\beta $ and a satisfy the following inequality:

$$ \begin{align*}\frac{a}{4 k} \frac{\binom{n}{k - 1}}{\binom{n}{p}}> kt \binom{n}{ k - p - 1} + 2^{ t + 1} t \beta a n^{ k - \ell - 1},\end{align*} $$

there is some edge E of $\mathcal {F}'\setminus \mathcal {Z}_{\mathcal {K}}(\varphi ^i(\mathcal {H}_i))$ that contains $\varphi ^i(F\cap E_{i+1})$ but does not intersect the rest of $\varphi ^i(\mathcal {H}_i)$ . Picking some $a_0$ sufficiently large in terms of $k, \ell , t$ and $\beta $ sufficiently small in terms of $k, \ell , t$ , the inequality holds and such E exists. We extend $\varphi ^i$ to an embedding $\varphi ^{i+1}$ of $\mathcal {H}_{i+1}$ into $\mathcal {F}'$ by mapping the vertices of $E_{i+1}\setminus (F\cap E_{i+1})$ injectively to the vertices of $E\setminus \varphi ^i (F\cap E_{i+1})$ so that $\varphi ^{i+1}(E_{i+1})=E$ . Note that $\varphi ^{i+1}(\mathcal {H}_{i+1})=\varphi ^i(\mathcal {H}_i)\cup \{E\}$ . Since $E \not \in \mathcal {Z}_{\mathcal {K}}(\varphi ^i(\mathcal {H}_i))$ and $\varphi ^{i + 1}(\mathcal {H}_{i+1})$ is admissible relative to $\mathcal {K}$ . This completes the induction. Now, $\varphi ^t(\mathcal {H}_t)$ is a copy of $\mathcal {H}$ that is admissible relative to $\mathcal {K}$ , completing our proof.

Proof. For convenience, let $\sigma =\sigma (\mathcal {H})$ . Let $\beta $ be sufficiently small and $a_0$ be sufficiently large. Let $\mathcal {K}$ be maximal collection of copies of $\mathcal {H}$ in $\mathcal {F}$ that satisfies condition 2. Since any collection consisting of a single copy of $\mathcal {H}$ satisfies condition 2, $\mathcal {K}$ exists. We show that $\mathcal {K}$ must also satisfy condition 1. Suppose on the contrary that $|\mathcal {K}| < \beta ^t n^{ k - 1} (n^{k - \ell })^{t - 1}$ . To derive a contradiction, as in the proof Lemma 5.2, it suffices to show that $\mathcal {F}$ contains a copy $\mathcal {H}^*$ of $\mathcal {H}$ that is admissible relative to $\mathcal {K}$ .

By definition, $\mathcal {K}$ is $\beta n^{ k- \ell - 1}$ -graded. Recall that a set $S \subseteq \mathcal {F}$ is saturated with respect to $\mathcal {K}$ if $\deg _{\mathcal {K}}(S) = \left \lfloor (\beta n^{k - \ell })^{t - |S|} \right \rfloor $ . Let $\mathcal {F}_{\mathrm {heavy}} := \{F \in \mathcal {F}: \deg _{\mathcal {K}}(F) = \left \lfloor (\beta n^{k - \ell })^{t - 1} \right \rfloor \}$ . Note that $\mathcal {F}_{\mathrm {heavy}}$ consists of all the saturated edges. By Lemma 5.1, $|\mathcal {F}_{\mathrm {heavy}}| \leq 2t \beta n^{ k - 1}<\frac {\gamma }{2} \binom {n}{k-1}$ , if $\beta $ is sufficiently small. Let $\widehat {\mathcal {F}} := \mathcal {F} - \mathcal {F}_{\mathrm {heavy}}$ . Then $|\widehat {\mathcal {F}}| \geq (\sigma -1 + \frac {\gamma }{2}) \binom {n}{ k - 1}.$ By definition, all edges of $\widehat {\mathcal {F}}$ are admissible relative to $\mathcal {K}$ . Let $s = v(\mathcal {H})$ . Now we apply Theorem 3.5 to $\widehat {\mathcal {F}}$ with

$$\begin{align*}a = (\sigma - 1 + \frac{\gamma}{2}) \text{ and }\varepsilon = \frac{1}{\sigma} \left(\frac{\gamma}{2}\right)^{ k - 2}. \end{align*}$$

There are two cases to consider:

Case 1. $\widehat {\mathcal {F}}$ satisfies case (1) of Theorem 3.5.

In this case, we find an $\mathcal {F}' \subseteq \widehat {\mathcal {F}}$ which is s-super-homogeneous, $|\mathcal {F}'|\geq \frac {c(k, s) \gamma ^{ k -2}}{2^{k - 1} \sigma } \binom {n}{ k - 1}$ , and $\mathrm {Int}(\mathcal {F}')$ is of type 1, where without loss of generality $2^{[ k - 2]} \subseteq \mathrm {Int}(\mathcal {F}')$ . Let $(X_1,\dots , X_k)$ be the k-partition associated with $\mathrm {Int}(\mathcal {F}')$ . Since $\mathcal {H}$ is $2$ -contractible, each edge contains at least two vertices of degree $1$ . We will designate two of them as expansion vertices for E.

Let $E_1,\cdots , E_t$ be a tree-defining ordering of the edges of $\mathcal {H}$ . For each $i\in [t]$ , let $\mathcal {H}_i=\{E_1,\dots , E_i\}$ . Let $\Pi _i$ denote the set of embeddings $\varphi ^i$ of $\mathcal {H}_i$ into $\mathcal {F}'$ satisfying that $\varphi ^i(\mathcal {H}_i)$ is admissible relative to $\mathcal {K}$ and $\varphi ^i$ maps all expansion vertices in $\mathcal {H}_i$ to $X_{k-1}\cup X_k$ . We use induction on i to prove that for each $i\in [t]$ , $\Pi _i\neq \emptyset $ .

For the base step, we let $F_1$ be any edge of $\mathcal {F}'$ . Let $\varphi ^1$ be a bijection that maps the vertices of $E_1$ to the vertices of $F_1$ such that the two expansion vertices are mapped to $F_1\cap X_{k-1}$ and $F_1\cap X_k$ , respectively. Since all edges of $\mathcal {F}'$ are admissible, $\varphi ^1(\mathcal {H}_1)$ is admissible. Hence, $\Pi _1\neq \emptyset $ .

For the induction step, let $1\leq i\leq t-1$ and suppose that $\Pi _i\neq \emptyset $ . Let $\varphi ^i\in \Pi _i$ . Let F be a parent edge of $E_{i+1}$ in $\mathcal {H}_i$ . Since vertices in $F\cap E_{i+1}$ have degree at least two in $\mathcal {H}$ , they are not expansion vertices in $\mathcal {H}_i$ . Hence, $\varphi ^i(F\cap E_{i+1})\subseteq \bigcup _{j=1}^{k-2} X_j$ . Since $2^{[k-2]}\subseteq \mathrm {Int}(\mathcal {F})$ , we have $\pi (\varphi ^i(F \cap E_{i+1})) \in \mathrm {Int}(\mathcal {F})$ . So, in particular that $\mathcal {L}_{\mathcal {F}}(\varphi ^i(F \cap E_{i+1}))$ is s-diverse. Therefore, we have that at least $1 - \frac {v(H_i)}{s} \geq \frac {1}{s}$ proportion of the edges in $\mathcal {L}_{\mathcal {F}}( \varphi ^i(F \cap E_i))$ do not intersect the rest of $\varphi ^i(\mathcal {H}_i)$ . Furthermore, by Lemma 5.1, with $d=\beta n^{k-\ell -1}$ , $|\mathcal {Z}_{\mathcal {K}}(\varphi ^i(\mathcal {H}_i))| \leq t 2^{ t + 1} \beta n^{ k - \ell - 1}$ . Let $p=|F\cap E_{i+1}|$ . Since $\mathcal {H}$ is $\ell $ -overlapping, $p\leq \ell $ . Since $\mathcal {F}'$ is s-super-homogoneus and $\pi (F\cap E_{i+1})\in \mathrm {Int}(\mathcal {F}')$ ,

$$\begin{align*}|\mathcal{L}_{\mathcal{F}'}(\varphi^i(F \cap E_{i+1}))|\geq \frac{|\mathcal{F}'|}{2kn^p} \geq \frac{c(k,s) \gamma^{k - 2} }{2^{k} \sigma k!} n^{ k - p - 1 } \geq \frac{c(k,s) \gamma^{k - 2}}{2^{k} \sigma k!} n^{ k - \ell - 1 }.\end{align*}$$

By picking $\beta $ sufficiently small in terms of $\gamma $ and $\mathcal {H}$ , we can ensure that

$$\begin{align*}\frac{1}{s} | \mathcal{L}_{\mathcal{F}}'( \varphi^i(F \cap E_{i+1}))|> |\mathcal{Z}_{\mathcal{K}}(\varphi^i(\mathcal{H}_i))|.\end{align*}$$

Hence, $\mathcal {F}'\setminus \mathcal {Z}(\varphi ^i(\mathcal {H}_i))$ contains an edge E that contains $\varphi ^i(F\cap E_{i+1})$ but does not intersect the rest of $\varphi ^i(\mathcal {H}_i)$ . We extend $\varphi ^i$ to an embedding of $\mathcal {H}_{i+1}$ in $\mathcal {F}'$ by mapping the vertices of $E_{i+1}\setminus (F\cap E_{i+1})$ injectively into $E\setminus \varphi ^i(F\cap E_{i+1})$ , so that $\varphi ^{i+1}(E_{i+1})=E$ and the expansion vertices of E are mapped into $X_{k - 1}$ and $X_k$ . Since $E \not \in \mathcal {Z}_{\mathcal {K}}(\varphi ^i(\mathcal {H}_i))$ , $\varphi ^{i + 1}(\mathcal {H}_{i + 1})$ is admissible relative to $\mathcal {K}$ . This completes the induction. Now, $\varphi ^t(\mathcal {H}_t)$ is a copy of $\mathcal {H}$ in $\mathcal {F}'$ that is admissible relative to $\mathcal {K}$ . This completes the proof for Case 1.

Case 2. $\widehat {\mathcal {F}}$ satisfies case (2) of Theorem 3.5.

In this case, there exist a $W \subseteq [n]$ and $\mathcal {F}" \subseteq \widehat {\mathcal {F}}$ satisfying

1. (a) $|W| \leq \left ( \frac {5\cdot 2^{2k - 2} \sigma ^2}{c(k,s) \gamma ^{2k - 4}} \right )^{k - 1}.$

2. (b) $|\mathcal {F}"| \geq (1-\frac {\gamma ^{k - 2}}{ 2^{ k - 2} \cdot \sigma } )|\widehat {\mathcal {F}}|\geq (1-\frac {\gamma }{4\sigma })(\sigma -1+\frac {\gamma }{2}) \binom {n}{k-1} \geq (\sigma - 1 + \frac {\gamma }{8} ) \binom {n}{k - 1} .$

3. (c) Every $F \in \mathcal {F}"$ satisfies $|W \cap F| = 1$ .

For each $S\in \binom {W}{\sigma }$ , let $\mathcal {L}^*(S) = \bigcap _{v \in S} \mathcal {L}_{\mathcal {F}"}(v)$ . Let $\mathcal {D}$ denote the set of $(k-1)$ -sets D in $[n]\setminus W$ with $d_{\mathcal {F}"}(D)\geq \sigma $ . Since each edge of $\mathcal {F}"$ contains one vertex in W and a $(k-1)$ -set in $[n]\setminus W$ , via double counting, we get

$$\begin{align*}\sum_{S \in \binom{W}{\sigma}} |\mathcal{L}^*(S)| = \sum_{D \in \mathcal{D}} \binom{d_{\mathcal{F}"}(D)}{\sigma}.\end{align*}$$

By our assumption, $|\mathcal {F}"|\geq (\sigma -1+\frac {\gamma }{8})\binom {n}{k-1}$ . This implies that $m:=\sum _{D\in \mathcal {D}} d_{\mathcal {F}"} (D)\geq \frac {\gamma }{8}\binom {n}{k-1}$ . Thus, by observing that if $d_{\mathcal {F}"}(D) \geq \sigma $ , $\binom {d_{\mathcal {F}"}(D)}{\sigma } \geq \frac {d_{\mathcal {F}"}(D)}{\sigma }$ , the right-hand side is at least $\frac {\gamma }{8 \sigma } \binom {n}{k -1}$ .

Hence, there exists a $S \in \binom {W}{\sigma }$ , such that

(12) $$ \begin{align} |\mathcal{L}^*(S)| \geq\frac{\gamma}{8 \sigma |W|^\sigma} \binom{n}{ k - 1}. \end{align} $$

We fix such an S. By Lemma 4.1, $\mathcal {L}^*(S)$ contains a subgraph $\mathcal {L}'$ such that $|\mathcal {L}'|\geq \frac {1}{2}|\mathcal {L}^*(S)|$ and for each $i\in [k-1]$ , $\delta _i(\mathcal {L}')\geq \frac {1}{2k}\frac {|\mathcal {L}^*(S)|}{\binom {n}{i}}$ . Note that since $\mathcal {L}'\subseteq \mathcal {L}^*(S)$ , for each $D\in \mathcal {L}'$ and each $v\in S$ , $D\cup \{v\}\in \mathcal {F}"$ .

Let $E_1,\dots , E_t$ be a tree-defining order of $\mathcal {H}$ . For each $i\in [t]$ , let $\mathcal {H}_i=\{E_1,\dots , E_i\}$ . Let R be a minimum cross-cut of $\mathcal {H}$ . For each $i\in [t]$ , let $\{v_i\}=E_i\cap R$ and $E^{\prime }_i=E_i\setminus \{v_i\}$ . Let $\mathcal {H}' = \mathcal {H} - R$ . It is easy to see by definition that $\mathcal {H}'$ is a $(k-1)$ -tree for which $E^{\prime }_1,\dots , E^{\prime }_t$ is a tree-defining ordering of $\mathcal {H}'$ . Furthermore, since each $E_i$ contains two degree $1$ vertices, $E^{\prime }_1,\dots , E^{\prime }_t$ are all distinct.

For each $i\in [t]$ , let $\Pi _i$ denote the collection of embeddings of $\varphi ^i$ of $\mathcal {H}_i$ into $\mathcal {F}'$ such that $\varphi (\mathcal {H}_i)$ is admissible relative to $\mathcal {K}$ . We prove by induction on i that $\Pi _i\neq \emptyset $ . Let g be any bijection from R to S. For the base step, let $D_1$ be any edge of $\mathcal {L}'$ . We define $\varphi ^1$ to be any mapping from $V(\mathcal {H}_1)$ to $V(\mathcal {F}")$ that maps $v_1$ to $g(v_1)$ and $E^{\prime }_1$ to $D_1$ . By earlier discussion, $\varphi ^1(E_1)= D_1\cup \{v_1\}$ is an edge in $\mathcal {F}"$ . Since all edges in $\widehat {\mathcal {F}}$ are admissible, all edges in $\mathcal {F}"$ are admissible. Hence, $\varphi ^1(\mathcal {H}_1)= \varphi ^1(E_1)$ is admissible relative to $\mathcal {K}$ . For the induction step, let $1\leq i\leq t-1$ and suppose $\Pi _i\neq \emptyset $ . Let $\varphi ^i\in \Pi _i$ . Suppose a parent edge of $E^{\prime }_{i+1}$ in $\mathcal {H}^{\prime }_i$ is $E^{\prime }_j$ , for some $j\leq i$ . Let $p=|E_j\cap E^{\prime }_{i+1}|$ . Since $\mathcal {H}$ is $\ell $ -overlapping, $\mathcal {H}'$ is $\ell $ -overlapping. So $p\leq \ell $ . By (12) and subsequent discussion, $d_{\mathcal {L}'}(\varphi ^i(E^{\prime }_j\cap E^{\prime }_{i+1}))\geq \frac {\gamma }{16\sigma |W|^\sigma }\frac {\binom {n}{k-1}}{\binom {n}{p}}$ . The number of edges of $\mathcal {L}'$ containing $\varphi ^i(E^{\prime }_j\cap E^{\prime }_{i + 1})$ and intersect the rest of $\varphi ^i(\mathcal {H}_i)$ is at most $kt \binom {n}{ k - p - 2}$ . Furthermore, by Lemma 5.1, with $d=\beta n^{k-\ell -1}$ , $|\mathcal {Z}_{\mathcal {K}}(\varphi ^i(\mathcal {H}_i))|\leq 2^{ t + 1} t \beta n^{ k - \ell -1}$ . Let $\mathcal {Z}'=\{E\setminus R: E\in \mathcal {Z}_{\mathcal {K}}(\varphi ^i(\mathcal {H}_i))\}$ . Then $|\mathcal {Z}'|\leq 2^{t+1}t\beta n^{k-\ell -1}$ . By choosing $\beta $ small enough, for sufficiently large n, we have

$$\begin{align*}\frac{\gamma}{16 k \sigma |W|^\sigma } \frac{\binom{n}{k - 1}}{\binom{n}{p}}> kt \binom{n}{ k - p - 2} + 2^{ t + 1} t \beta n^{ k - \ell -1},\end{align*}$$

where we used the fact that $|W|$ is bounded by a constant.

Hence, $\mathcal {L}'\setminus Z'$ contains an edge D that contains $\varphi ^i(E_j'\cap E_{i+1}')$ but does not intersect the rest of $\varphi ^i(\mathcal {H}_i)$ . We extend $\varphi ^i$ to an embedding of $\mathcal {H}_{i+1}$ in $\mathcal {F}'$ by mapping the vertices of $E^{\prime }_{i+1}\setminus (E^{\prime }_j\cap E^{\prime }_{i+1})$ injectively into $D\setminus \varphi ^i(E^{\prime }_j\cap E^{\prime }_{i+1})$ and mapping $v_{i+1}$ to $g(v_{i + 1})$ so that $\varphi ^{i+1}(E_{i+1})=D \cup \{ g(v_{i + 1})\}$ . Since $D \not \in \mathcal {Z}'$ , $\varphi ^{i+1}(E_{i+1})\notin \mathcal {Z}_{\mathcal {K}}(\varphi ^i(\mathcal {H}_i))$ , and hence $\varphi ^{i + 1}(\mathcal {H}_{i + 1})$ is admissible relative to $\mathcal {K}$ . This completes the induction. Now, $\varphi ^t(\mathcal {H}_t)$ is a copy of $\mathcal {H}$ in $\mathcal {F}"$ that is admissible relative to $\mathcal {K}$ . This completes the proof for Case 2.

Proof. Given $\mathcal {F}$ , let $\mathcal {K}$ be the collection of copies of $\mathcal {H}$ returned by Theorem 5.4. We will view $\mathcal {K}$ as a t-uniform hypergraph with $V(\mathcal {K})=E(\mathcal {F})$ whose hyperedges are the copies of $\mathcal {H}$ in the collection. By the conditon on $\mathcal {K}$ , we have that for all $1\leq b\leq t$

$$ \begin{align*}\Delta_{b}(\mathcal{K}) \leq c \tau(\mathcal{F})^{b - 1}\frac{|\mathcal{K}|}{|\mathcal{F}|}\end{align*} $$

where $\tau (\mathcal {F}) := \frac {1}{n^{k - \ell - 1}} \frac {\binom {n}{k - 1}}{|\mathcal {F}|}$ .

Then, applying Lemma 6.1, there is a $\delta $ and a collection $\mathcal {C} \subseteq 2^{E(\mathcal {F})}$ satisfying:

1. 1. For every independent set $I \subseteq \mathcal {H}$ , there is a $C \in \mathcal {C}$ such that $I \subseteq C$ .

2. 2. $|\mathcal {C}| \leq \binom {n}{\leq \frac {t}{n^{k - \ell - 1}} \binom {n}{ k - 1} }$ .

3. 3. For every $C \in \mathcal {C}$ , $|V(C)| \leq (1 - \delta )|\mathcal {F}|$ .

Noting that every $\mathcal {H}$ -free subfamily of $\mathcal {F}$ corresponds to an independent set in $\mathcal {K}$ , the result follows.

Proof. Let $\delta := \delta (\varepsilon , \mathcal {H})$ be the constant given by Lemma 6.2 and choose K large enough depending on $\varepsilon , \delta , t$ . Let n be sufficiently large.

We will build a sequence of auxiliary rooted trees $\mathcal {A}_1, \dots , \mathcal {A}_m$ such that leaves of $\mathcal {A}_m$ will be our containers. We begin by fixing $\mathcal {A}_1$ to be the tree with a single vertex, the root $K_n^{(k)}$ . To construct $\mathcal {A}_2$ , we apply Lemma 6.2 to $K_n^{(k)}$ to find a collection $\mathcal {C}$ of subgraphs of $K_n^{(k)}$ . We form $\mathcal {A}_2$ by adding each container $C \in \mathcal {C}$ as a vertex and connect it to the root $K_n^{(k)}$ . Observe that in $\mathcal {A}_2$ , $K_n^{(k)}$ has no more than