Proof. Suppose to the contrary that there exists a non-trivial intersecting subgraph $\mathcal{F} \subset \widehat{\mathcal{S}}$ with $3m+1 \ge 11$ edges. It follows from Theorem 2.1 that $\mathcal{F}$ is contained in $H_i(n)$ for some $i\in \{0,\ldots, 5\}$ . Notice that $\Delta _2(\mathcal{F}) \le \Delta _2(\widehat{\mathcal{S}}) = m$ . If $\mathcal{F}$ is contained in $H_i(n)$ for some $i\in \{0,\ldots, 5\}\setminus \{1\}$ , then by Observation 2.2, we would have $\Delta _2(\mathcal{F}) \ge \min \left \{\lceil \frac{3m+1}{3} \rceil, \lceil \frac{3}{2}m-1 \rceil, 3m-5\right \} \gt m$ , a contradiction. Therefore, $\mathcal{F} \subset H_1(n)$ . Then it is easy to see from the structure of $H_1(n)$ that $\mathcal{F}$ contains four vertices $v_1,v_2,v_3,v_4$ such that

\begin{equation*}\deg _{\widehat {\mathcal {S}}}(v_1v_2) + \deg _{\widehat {\mathcal {S}}}(v_1v_3) + \deg _{\widehat {\mathcal {S}}}(v_1v_4) \ge 3m,\end{equation*}

This implies that $\deg _{\widehat{\mathcal{S}}}(v_1v_2) = \deg _{\widehat{\mathcal{S}}}(v_1v_3) = \deg _{\widehat{\mathcal{S}}}(v_1v_4) = m$ since each of them is at most $m$ . However, this is impossible since the set $\left \{ \{u,v\} \subset V(\mathcal{S})\colon \deg _{\widehat{S}}(uv)=m \right \}$ consists of $n/3$ copies of pairwise vertex-disjoint triangles.

Proof. Since $r(\mathcal{J}) = k$ and $\mathcal{J}$ is closed under intersection, we have $\mathcal{J} = 2^{[k]}\setminus \{[k]\}$ . Let $E\in \mathcal{H}$ and let $\bigcup _{i=1}^{p}A_i = E$ be an $\vec{a}$ -partition of $E$ . Since $\Pi (\mathcal{I}(E,\mathcal{H})) = \mathcal{J} =2^{[k]}\setminus \{[k]\}$ , we have $E\setminus A_i \in \mathcal{I}(E,\mathcal{H})$ for all $i \in [p]$ . Since $\mathcal{H}$ is $s$ -homogeneous, for every $i \in [p]$ there exists a $\Delta$ -system $\mathcal{D}_i$ of size $s$ with centre $E\setminus A_i$ . Since $s \ge kd+1$ , it follows from Lemma 3.5 that $\mathcal{H}$ contains an $(\vec{a},d)$ - $\Delta$ -system.

Proof. Let $s = kd+1$ and $\mathcal{H}^{\ast }$ be a maximum $s$ -homogeneous subgraph of $\mathcal{H}$ with intersection pattern $\mathcal{J}$ . Then by Theorem 3.7, $|\mathcal{H}^{\ast }| \ge c(k,d) |\mathcal{H}|$ for some constant $c(k,d) \gt 0$ . Since $\mathcal{H}^{\ast }$ contains no $(\vec{a},d)$ - $\Delta$ -systems, it follows from Lemma 4.1 that $r(\mathcal{J}) \le k-1$ . Then it follows from Lemma 3.8 that $|\mathcal{H}^{\ast }| \le |\partial \mathcal{H}^{\ast }|$ . Therefore,

\begin{align*}|\partial \mathcal {H}| \ge |\partial \mathcal {H}^{\ast }| \ge |\mathcal {H}^{\ast }| \ge c(k,d)|\mathcal {H}|.\\[-33pt]\end{align*}

Proof. Without loss of generality, we may assume that $i_0=1$ . By assumption, $E_0\setminus A_i$ is the centre of a $\Delta$ -system of size $s\ge kd +1$ in $\mathcal{H}^{\ast }\subset \mathcal{H}$ for $2 \le i \le k$ , and $E_0\setminus A_1$ is the centre of a $\Delta$ -system of size $2$ in $\mathcal{H}$ , i.e. $\{E_0, E\}$ . So it follows from Lemma 3.5 that $\mathcal{H}$ contains an $(\vec{a},d)$ - $\Delta$ -system.

Proof. Let $E = \{v_1,\ldots,v_k\} \in \mathcal{H}^{\ast }$ and suppose that $v_1$ is contained in all $(k-1)$ -sets in $\mathcal{I}(E,\mathcal{H}^{\ast })$ . Note that since $\mathcal{I}(E,\mathcal{H}^{\ast })$ is closed under intersection, all proper subsets of $E$ that contain $v_1$ are contained in $E$ . Let $F \in \mathcal{H}$ and suppose that $|E \cap F| = k-t$ for some $1 \le t \le a_1$ . Suppose to the contrary that $v_1\not \in F$ .

If $t = a_1$ , then let $\bigcup _{i\in [p]}A_i = E$ be an $\vec{a}$ -partition such that $A_1 = E\setminus F$ . For $2\le i \le p$ since $v_1\in E\setminus A_i$ , we have and it follows from the assumption that $E\setminus A_i$ is the centre of a $\Delta$ -system of size $s$ in $\mathcal{H}^{\ast }$ . On the other hand, notice that $E\setminus A_1$ is the centre of a $\Delta$ -system of size $2$ , i.e. $\{E,F\}$ . So it follows from Lemma 3.5 that $\mathcal{H}$ contains an $(\vec{a},d)$ - $\Delta$ -system, a contradiction. Therefore, $t \lt a_1$ .

Let $M\subset E$ be a set of size $k-a_1+t$ such that $E\setminus F\subset M$ . Since $|M|\le k-1$ and $v_1 \in M$ , we have $M\in \mathcal{I}(E,\mathcal{H}^{\ast })$ . By assumption, $M$ is the center of a $\Delta$ -system $\mathcal{D}\subset \mathcal{H}^{\ast }$ of size $s$ in $\mathcal{H}^{\ast }$ . Since $F\setminus E$ can have nonempty intersection with at most $k\lt s$ edges in $\mathcal{D}$ , there exists an edge $E_1\in \mathcal{D}$ such that $(E_1\setminus E)\cap F = \emptyset$ . Notice that $E_1$ satisfies $E_1\cap F= M\setminus (E\setminus F)$ and $|E_1\cap F| = k-a_1$ . Since $\Pi (\mathcal{I}(E_1,\mathcal{H}^{\ast })) = \Pi (\mathcal{I}(E,\mathcal{H}^{\ast }))$ , applying the same argument as above to $E_1$ and $F$ we obtain that $\mathcal{H}$ contains an $(\vec{a},d)$ - $\Delta$ -system, a contradiction.

Proof. Suppose to the contrary that this statement is not true. Without loss of generality, we may assume that $I = [t]\setminus \{1\}$ and there exist distinct vertices $u_2,\ldots,u_t$ such that $F_i:= (E\setminus \{v_i\})\cup \{u_i\} \in \mathcal{H}$ for $2 \le i \le t$ .

By assumption $\mathcal{I}(E,\mathcal{H}^{\ast })$ contains all $(k-1)$ -sets that contain $\{v_1,\ldots,v_t\}$ . Since $\mathcal{I}(E,\mathcal{H}^{\ast })$ is closed under intersection, $\mathcal{I}(E,\mathcal{H}^{\ast })$ contains all proper subsets of $E$ that contain $\{v_1,\ldots,v_t\}$ . In other words, if $A\subset \{v_{t+1},\ldots,v_k\}$ and $A\neq \emptyset$ , then $E\setminus A \in \mathcal{I}(E,\mathcal{H}^{\ast })$ .

We claim that for all $\{i,j\}\subset [t]$ we have $E\setminus \{v_i,v_j\} \in \mathcal{I}(E,\mathcal{H}^{\ast })$ . This is true because otherwise it would follow from the assumption $E\setminus \{v_i\}, E\setminus \{v_j\} \not \in \mathcal{I}(E,\mathcal{H}^{\ast })$ that $r(\mathcal{I}(E,\mathcal{H}^{\ast })) \le k-2$ , which contradicts the fact that $r(\mathcal{I}(E,\mathcal{H}^{\ast })) = k-1$ . Now by the argument above and the property that $\mathcal{I}(E,\mathcal{H}^{\ast })$ is closed under intersection, we know that if a set $A\subset E$ satisfies $|A \cap \{v_1,\ldots,v_t\}| \ge 2$ , then $E\setminus A \in \mathcal{I}(E,\mathcal{H}^{\ast })$ .

Let $i_0 \in [p]$ such that $\sum _{i=1}^{i_0-1}a_i \lt t \le \sum _{i=1}^{i_0}a_i$ , and let $\ell = t - \sum _{i=1}^{i_0-1}a_i$ . Note that $i_0 \le t$ . Recall that $a_1 \ge \cdots \ge a_p\ge 1$ and $a_1 \ge 2$ .

Suppose that $\ell \ge 2$ . Then there exists an $\vec{a}$ -partition $E = \bigcup _{i\in [p]}A_i$ such that $A_1, \ldots, A_{i_0-1} \subset \{v_1, \ldots, v_t\}$ , $|A_{i_0} \cap \{v_1, \ldots, v_t\}| = \ell \ge 2$ , and $A_{i_0+1},\ldots, A_p \subset \{v_{t+1},\ldots, v_k\}$ . Since $a_1 \ge \cdots \ge a_{i_0-1} \ge a_{i_0} \ge 2$ , by the argument above, $E\setminus A_{i} \in \mathcal{I}(E,\mathcal{H}^{\ast })$ for all $i\in [p]$ . Therefore, $E\setminus A_{i}$ is the centre of a $\Delta$ -system of size $s \ge kd +1$ in $\mathcal{H}^{\ast }$ for $i\in [p]$ , and it follows from Lemma 3.5 that $\mathcal{H}$ contains an $(\vec{a},d)$ - $\Delta$ -system, a contradiction. Therefore, $\ell = 1$ .

Suppose that $a_{i_0} = 1$ . Then we let $E = \bigcup _{i\in [p]}A_i$ be an $\vec{a}$ -partition such that $\bigcup _{i \in [i_0]}A_i = \{v_1,\ldots, v_t\}$ and $v_1 \in A_1$ . Since $A_1 \subset \{v_1,\ldots, v_t\}$ and $|A_1| \ge 2$ , we have $E\setminus A_1 \in \mathcal{I}(E,\mathcal{H}^{\ast })$ . So $E\setminus A_1$ is the centre of a $\Delta$ -system $\mathcal{D}_1\subset \mathcal{H}^{\ast }$ of size $s$ . We will assume that $a_2 = \cdots = a_{i_0} = 1$ . Other cases follow analogously. For $i_0+1 \le i \le p$ since $A_{i}\subset \{v_{t+1},\ldots,v_k\}$ , we have $E\setminus A_{i} \in \mathcal{I}(E,\mathcal{H}^{\ast })$ and $E\setminus A_{i}$ is the centre of a $\Delta$ -system $\mathcal{D}_i\subset \mathcal{H}^{\ast }$ of size $s$ . Notice that by assumption, for every $2 \le i \le i_0$ we have $F_{i} \in \mathcal{H}$ such that $F_{i} \cap E = E\setminus \{v_{i}\}$ , and moreover, sets in $\{F_{i} \setminus E\colon 2 \le i \le i_0\}$ are disjoint. Now, fix $F_{i}$ for $2\le i \le i_0$ and choose greedily a set $E_j \in \mathcal{D}_j$ for $i_0\lt j \le p$ such that sets in $\{F_{i} \setminus E\colon 2 \le i \le i_0\} \cup \{E_j\setminus E\colon i_0\lt j \le p\}$ are disjoint (since $s\ge kd+1$ such $E_j$ ’s always exist). Finally, choose $d-p+1$ sets from $\mathcal{D}_1$ such that they have empty intersection with all sets in $\{F_{i} \setminus E\colon 2 \le i \le i_0\} \cup \{E_j\setminus E\colon i_0\lt j \le p\}$ (since $s\ge kd+1$ we can always find such $d-p+1$ sets). This shows that $\mathcal{H}$ contains an $(\vec{a},d)$ - $\Delta$ -system, a contradiction. Therefore, $a_{i_0} \ge 2$ .

Suppose that $a_1 \ge 3$ . Then we let $E = \bigcup _{i\in [p]}A_i$ be an $\vec{a}$ -partition such that $A_2 \cup \cdots \cup A_{i_0-1} \subset \{v_1,\ldots, v_t\}$ , $v_{t+1}\in A_1$ and $A_1\setminus \{v_{t+1}\}\subset \{v_1,\ldots, v_t\}$ , and $\{v_1,\ldots, v_t\} \setminus \left (\bigcup _{i\in [i_0-1]}A_i\right ) \subset A_{i_0}$ . By the argument in the last paragraph, we have $|A_i \cap \{v_1,\ldots, v_t\}| \ge 2$ for all $i \in [i_0]$ . The partition $E = \bigcup _{i\in [p]}A_i$ we chose ensures that $A_{j} \subset \{v_{t+1},\ldots,v_k\}$ for all $i_0+1 \le j \le p$ . Therefore, for every $i\in [p]$ the set $E\setminus A_{i}$ is contained in $\mathcal{I}(E,\mathcal{H}^{\ast })$ (this follows from the second and the third paragraphs in the proof of Claim 4.6), and hence, $E\setminus A_{i}$ is the center of a $\Delta$ -system of size $s \ge kd +1$ in $\mathcal{H}^{\ast }$ . So, it follows from Lemma 3.5 that $\mathcal{H}$ contains an $(\vec{a},d)$ - $\Delta$ -system, a contradiction. Therefore, $a_1 = 2$ .

Suppose that $a_{p} = 1$ . Then we let $E = \bigcup _{i\in [p]}A_i$ be an $\vec{a}$ -partition such that $A_1 \cup \cdots \cup A_{i_0-1} = \{v_1,\ldots, v_{t-1}\}$ and $A_{p} = \{v_t\}$ . For every $i\in [i_0-1]$ since $|A_i\cap E| \ge 2$ , we have $E\setminus A_{i} \in \mathcal{I}(E,\mathcal{H}^{\ast })$ . For every $i_0\le j \le p-1$ since $A_j \subset \{v_{t+1},\ldots, v_k\}$ , we have $E\setminus A_{j} \in \mathcal{I}(E,\mathcal{H}^{\ast })$ . Therefore, $E\setminus A_{i}$ is the center of a $\Delta$ -system of size $s \ge kd +1$ in $\mathcal{H}^{\ast }$ for $i\in [p-1]$ . Note that $E\setminus A_p$ is the center of a $\Delta$ -system of size $2$ in $\mathcal{H}$ , i.e. $\{E,F_{t}\}$ . So it follows from Lemma 3.5 that $\mathcal{H}$ contains an $(\vec{a},d)$ - $\Delta$ -system, a contradiction. Therefore, $a_p = 2$ .

Now we have $a_1 = \cdots = a_p =2$ and $\ell = 1$ . This implies that $t$ is odd, $t\ge 3$ , $k$ is even, and $k\gt t$ . Since $E\setminus \{v_k\} \in \mathcal{I}(E,\mathcal{H}^{\ast })$ , we know that $E\setminus \{v_k\}$ is the center of a $\Delta$ -system $\mathcal{D}$ of size $s$ in $\mathcal{H}^{\ast }$ . So we can choose a set $F_k$ from $\mathcal{D}$ such that $F_k\setminus E$ has empty intersection with $\{u_2,\ldots,u_t\}$ . In other words, there exists a vertex $u_{k}$ that is not contained in $\{u_2,\ldots,u_t\}$ such that $F_k = (E\setminus \{v_k\})\cup \{u_{k}\}$ . Let $F_k = \bigcup _{i\in [p]}A_i$ be an $\vec{a}$ -partition such that $A_1 = \{v_2,u_k\}$ , $A_2 = \{v_1,v_3\}$ , and $A_{i} = \{v_{2i-2},v_{2i-1}\}$ for $3\le i \le p$ . Then for every $i \in [p]\setminus \{1\}$ , either $A_i \subset \{v_1,\ldots,v_t\}$ or $A_i \subset \{v_{t+1},\ldots,v_{k-1}\}$ . Since $\Pi (\mathcal{I}(F_k,\mathcal{H}^{\ast })) = \Pi (\mathcal{I}(E,\mathcal{H}^{\ast }))$ , the set $F_k\setminus A_{i}$ is the center of a $\Delta$ -system of size $s \ge kd +1$ in $\mathcal{H}^{\ast }$ for $i \in [p]\setminus \{1\}$ . Since $F_k\setminus A_1$ is the center of a $\Delta$ -system of size $2$ in $\mathcal{H}$ , i.e. $\{F_k,F_2\}$ . So it follows from Lemma 3.5 that $\mathcal{H}$ contains an $(\vec{a},d)$ - $\Delta$ -system, a contradiction. This completes the proof of Claim 4.6.

Proof. First, we show that $|\widehat{\mathcal{H}}_{3}| = O(n^{k-2})$ . We may assume that $\widehat{\mathcal{H}}_{3} \neq \emptyset$ . Recall that $\mathcal{H}_{m+1}$ is a maximum $s$ -homogeneous subgraph of $\widehat{\mathcal{H}}_{3}$ with intersection pattern $\mathcal{J}_{m+1}$ of rank at most $k-2$ . It follows from Theorem 3.7 that there exists a constant $c(k,s)\gt 0$ such that $|\mathcal{H}_{m+1}| \ge c(k,s)|\widehat{\mathcal{H}}_{3}|$ . Since $r(\mathcal{J}_{m+1}) \le k-2$ , it follows from Lemma 3.8 that $|\mathcal{H}_{m+1}| \le |\partial _2\mathcal{H}_{m+1}| \le \binom{n}{k-2}$ . Therefore, $|\widehat{\mathcal{H}}_{3}| \le \frac{1}{c(k,s)}|\mathcal{H}_{m+1}|\le \frac{1}{c(k,s)}\binom{n}{k-2}$ .

Next we show that $\widehat{\mathcal{H}}_{2} \le 2 \epsilon k \binom{n-1}{k-1}$ . By Equation (1) and Lemma 4.5, we have

\begin{align*} |\partial \mathcal{H}| = \sum _{E\in \mathcal{H}}\omega _{\mathcal{H}}(E) \ge \sum _{E\in \widehat{\mathcal{H}}_{1}}\omega _{\mathcal{H}}(E) + \sum _{E\in \widehat{\mathcal{H}}_{2}}\omega _{\mathcal{H}}(E) \ge |\widehat{\mathcal{H}}_{1}| + \frac{k}{k-1}|\widehat{\mathcal{H}}_{2}|. \end{align*}

This implies that $|\widehat{\mathcal{H}}_{1}| + \frac{k}{k-1}|\widehat{\mathcal{H}}_{2}| \le \binom{n}{k-1}$ , which further gives

\begin{align*} |\widehat{\mathcal{H}}_{2}| & = (k-1)\left (|\widehat{\mathcal{H}}_{1}| + \frac{k}{k-1}|\widehat{\mathcal{H}}_{2}| + |\widehat{\mathcal{H}}_{3}| - |\mathcal{H}|\right ) \\ & \le (k-1)\left (\binom{n}{k-1} + |\widehat{\mathcal{H}}_{3}| - (1-\epsilon )\binom{n-1}{k-1} \right ) \lt 2 \epsilon k \binom{n-1}{k-1}. \end{align*}

This completes the proof of Claim 5.1.

Proof. Suppose to the contrary that this is not true. Without loss of generality, we may assume that there exists $A \in \partial \mathcal{G}_1^{\prime} \cap \partial \mathcal{G}_{2}^{\prime}$ . Then this means that there exists $E_1 \in \mathcal{G}_1$ and $E_2 \in \mathcal{G}_2$ such that $E_1 = \{1,u\}\cup A$ and $E_2 = \{2,v\} \cup A$ for some $u,v \in [n]$ . Since $\mathcal{G}_1$ is $s$ -homogeneous and $|E_2\cap E_1| \ge k-2 \ge k-a_1$ , it follows from Lemma 4.4 that $1 \in E_2$ . Similarly, we obtain $2 \in E_1$ . Therefore, $E_1 = E_2 = \{1,2\} \cup A$ , which implies that $\{1,2\} \cup A \in \mathcal{G}_1\cap \mathcal{G}_2$ , a contradiction.

Proof. It is easy to see from the definition of $\mathcal{G}$ and $\mathcal{B}_0$ that $|E\cap B| \le k-a_1-1 \le k-3$ for all $E\in \mathcal{G}'$ and $B \in \mathcal{B}_0$ . Therefore, $\partial \mathcal{G}' \cap \partial _{2}\mathcal{B}_0 = \emptyset$ , and hence $|\partial \mathcal{G}'| + |\partial _{2}\mathcal{B}_0| \le \binom{n-1}{k-2}$ . Let $x \in \mathbb{R}^{+}$ be a positive integer such that $|\partial \mathcal{B}_0| = \binom{x}{k-1}$ . It follows from the Kruskal–Katona theorem and Proposition 4.2 that

\begin{align*} |\partial _2\mathcal{B}_0| \ge \frac{k-1}{x-k+1}|\partial \mathcal{B}_0| \ge \frac{k-1}{x-k+1}c(k,s)|\mathcal{B}_0|. \end{align*}

This together with Equation (4) and $|\partial \mathcal{G}'| + |\partial _{2}\mathcal{B}_0| \le \binom{n-1}{k-2}$ imply that

\begin{align*} \frac{k-1}{n-k+1}|\mathcal{G}'| + \frac{k-1}{x-k+1}c(k,s)|\mathcal{B}_0| \le \binom{n-1}{k-2}. \end{align*}

Consequently,

\begin{align*} |\mathcal{G}| + c(k,s)\frac{n-k+1}{x-k+1}|\mathcal{B}_0| & = |\mathcal{G}'| + c(k,s)\frac{n-k+1}{x-k+1}|\mathcal{B}_0| \\[3pt] & = \frac{n-k+1}{k-1}\left (\frac{k-1}{n-k+1}|\mathcal{G}'| + \frac{k-1}{x-k+1}c(k,s)|\mathcal{B}_0|\right ) \\[3pt] & \le \frac{n-k+1}{k-1}\binom{n-1}{k-2} = \binom{n-1}{k-1}. \end{align*}

By Theorem 3.4, we have $\binom{x}{k-1} = |\partial \mathcal{B}_0| \le k |\mathcal{B}_0| \le \delta k n^{k-1}$ . It follows that $x \le \left (k \delta \right )^{\frac{1}{k-1}} n \le c(k,s)n/100$ . Since $n \ge n_0(k,d,\delta )$ is large, we have

\begin{align*} c(k,s)\frac{n-k+1}{x-k+1} \ge c(k,s)\frac{n-k+1}{c(k,s)n/100 -k+1} \gt 4. \end{align*}

Therefore, the inequality above gives $|\mathcal{G}| + 4 |\mathcal{B}_0| \le \binom{n-1}{k-1}$ .

Proof. Suppose to the contrary that there exists an edge $E = \{v_1,\ldots,v_{k-1}\} \in \mathcal{B}^{\ast }_{1}$ such that for every $i\in [k-1]$ the set $E\setminus \{v_i\}$ is contained in some other set in $\mathcal{B}^{\ast }_{1} \cup \mathcal{G}'$ . Without loss of generality, we may assume that for some integer $\ell$ the set $E\setminus \{v_i\}$ is contained in $\mathcal{I}(E,\mathcal{G}')$ for all $i\in [\ell ]$ , and the set $E\setminus \{v_j\}$ is contained in $\mathcal{I}(E,\mathcal{B}^{\ast }_1)$ for all $\ell +1 \le j \le k-1$ .

Let $\mathcal{J}_{\mathcal{B}^{\ast }_{1}}$ be the intersection pattern of $\mathcal{B}^{\ast }_{1}$ . Let $\widehat{\mathcal{B}^{\ast }_{1}} = \{E\cup \{1\}\colon E\in \mathcal{B}^{\ast }_{1}\}$ , and notice that $\widehat{\mathcal{B}^{\ast }_{1}}$ is $s$ -homogeneous with intersection pattern $\mathcal{J}_{\widehat{\mathcal{B}^{\ast }_{1}}} := \{A\cup \{1\}\colon A\in \mathcal{J}_{\mathcal{B}^{\ast }_{1}}\}$ . Let $\widehat{E} = E\cup \{1\}$ and note that $\widehat{E}$ is an edge in $\widehat{\mathcal{B}^{\ast }_{1}}$ .

If $\ell = 0$ , then $\mathcal{J}_{\mathcal{B}^{\ast }_{1}} = 2^{[k-1]}\setminus \{[k-1]\}$ . Consequently, $r(\mathcal{I}(\widehat{E},\widehat{\mathcal{B}^{\ast }_{1}})) = k-1$ and $\mathcal{I}(\widehat{E},\widehat{\mathcal{B}^{\ast }_{1}})$ contains all $(k-1)$ -subsets of $\widehat{E}$ that contain $1$ . By the definition of $\mathcal{B}_1$ , there exists $B \in \mathcal{B}_0$ such that $|B\cap \widehat{E}|\ge k-a_1$ . It follows from Lemma 4.4 that $1 \in B$ , a contradiction. Therefore, $\ell \ge 1$ .

For every $i\in [\ell ]$ let $E_i \in \mathcal{G}$ be an edge such that $E_i \cap \widehat{E} = \widehat{E}\setminus \{v_i\}$ . Let $B \in \mathcal{B}_0$ be an edge such that $|B\cap \widehat{E}|\ge k-a_1$ . Suppose that $|B\cap \widehat{E}| = k-t$ , where $t$ is an integer that satisfies $1\le t \le a_1$ . Then for every $i\in [\ell ]$ we have $|B\cap E_i| \ge k-t-1$ . On the other hand, by the definition of $\mathcal{G}$ , for every $i\in [\ell ]$ we have $|B \cap E_i| \le k-a_1-1 \le k-t-1$ . Therefore, we must have $t = a_1$ , $|B\cap \widehat{E}| = k-a_1$ , and $v_i \in B$ for all $i\in [\ell ]$ . Let $\bigcup _{i\in [p]}A_i = \widehat{E}$ be an $\vec{a}$ -partition such that $A_1 = \widehat{E}\setminus B$ . Note that for $2\le i \le p$ , either there exists some $j\in [\ell ]$ such that $\widehat{E}\setminus A_i \subset E_{j}$ or $\{1,v_1, \ldots, v_{\ell }\} \subset \widehat{E}\setminus A_i$ . In the former case, since $1\in \widehat{E}\setminus A_i$ , it follows from the definition of $\mathcal{G}$ that $\widehat{E}\setminus A_i$ is the center of some $\Delta$ -system of size $s$ in $\mathcal{H}$ . In the latter case, we have $\widehat{E}\setminus A_i \in \mathcal{I}(\widehat{E},\widehat{\mathcal{B}^{\ast }_1})$ , which implies that $\widehat{E}\setminus A_i$ is the center of some $\Delta$ -system of size $s$ in $\widehat{\mathcal{B}^{\ast }_1}$ . Note that $E\setminus A_1$ is the center of a $\Delta$ -system of size $2$ , i.e. $\{\widehat{E},B\}$ . So, it follows from Lemma 3.5 that $\mathcal{H}$ contains an $(\vec{a},d)$ - $\Delta$ -system, a contradiction.

Proof. Suppose to the contrary that there exists an edge $E = \{v_1,\ldots,v_{k-1}\} \in \mathcal{B}^{\ast }_{2}$ such that for every $i\in [k-1]$ the set $E\setminus \{v_i\}$ is contained in some other set in $\mathcal{B}^{\ast }_{2} \cup \mathcal{G}'$ . Without loss of generality we may assume that there exists an integer $[\ell ]$ such that $E\setminus \{v_i\} \in \mathcal{I}(E,\mathcal{G}')$ for $i\in [\ell ]$ , and $E\setminus \{v_j\} \in \mathcal{I}(E,\mathcal{B}^{\ast }_2)$ for $\ell +1 \le j \le k-1$ .

Let $\mathcal{J}_{\mathcal{B}^{\ast }_{2}}$ be the intersection pattern of $\mathcal{B}^{\ast }_{2}$ . Let $\widehat{\mathcal{B}^{\ast }_{2}} = \{E\cup \{1\}\colon E\in \mathcal{B}^{\ast }_{2}\}$ . Note that $\widehat{\mathcal{B}^{\ast }_{2}}$ is also $s$ -homogeneous with intersection pattern $\mathcal{J}_{\widehat{\mathcal{B}^{\ast }_{2}}} := \{A\cup \{1\}\colon A\in \mathcal{J}_{\mathcal{B}^{\ast }_{2}}\}$ . Let $\widehat{E} = E\cup \{1\}$ and note that $\widehat{E}$ is an edge in $\widehat{\mathcal{B}^{\ast }_{2}}$ .

If $\ell = 0$ , then $\mathcal{J}_{\mathcal{B}^{\ast }_{2}} = 2^{[k-1]}\setminus \{[k-1]\}$ . Consequently, $r(\mathcal{I}(\widehat{E},\widehat{\mathcal{B}^{\ast }_{2}})) = k-1$ , and hence $\mathcal{I}(\widehat{E},\widehat{\mathcal{B}^{\ast }_{2}})$ contains all $(k-1)$ -subsets of $\widehat{E}$ that contain $1$ . Since $\mathcal{I}(\widehat{E},\widehat{\mathcal{B}^{\ast }_{2}})$ is closed under intersection, all proper subsets of $\widehat{E}$ that contain $1$ are contained in $\mathcal{I}(\widehat{E},\widehat{\mathcal{B}^{\ast }_{2}})$ . However, by the definition of $\mathcal{G}$ , this implies that $\widehat{E} \in \mathcal{G}$ , a contradiction. Therefore, $\ell \ge 1$ .

For every $i\in [\ell ]$ let $E_i \in \mathcal{G}$ be an edge such that $E_i \cap \widehat{E} = \widehat{E}\setminus \{v_i\}$ . Let $S$ be a proper subset of $\widehat{E}$ with $1\in S$ . If $v_i \not \in S$ for some $i\in [\ell ]$ , then $S\subset E_i$ and it follows from the definition of $\mathcal{G}$ that $S$ is the center of some $\Delta$ -system of size $s$ in $\mathcal{H}$ . If $\{v_1,\ldots, v_{\ell }\} \subset S$ , then $S \in \mathcal{I}(\widehat{E},\widehat{\mathcal{B}^{\ast }_{2}})$ and it follows that $S$ is the center of some $\Delta$ -system of size $s$ in $\mathcal{B}^{\ast }_2$ . Therefore, every proper subset of $\widehat{E}$ that contains $1$ is the center of some $\Delta$ -system of size $s$ in $\mathcal{H}$ . However, it follows from the definition of $\mathcal{G}$ that $\widehat{E} \in \mathcal{G}$ , a contradiction.