Proof. Let $H^{\prime}$ and $G^{\prime}$ be copies of $H$ and $G$ that are subgraphs of the $k$ th edge layers in hypercubes with ground sets $X$ and $Y$ , respectively, such that $V(H^{\prime}) \cap \binom{X}{k}$ and $V(H^{\prime}) \cap \binom{Y}{k}$ are the edge sets of $k$ -partite hypergraphs with parts $U_1,\ldots,U_k$ and parts $W_1,\ldots, W_k$ , respectively. Let $u$ and $w$ be respective copies of $v$ in $H^{\prime}$ and $G^{\prime}$ . We shall specify $X$ and $Y$ in two cases below.

Assume first that $v$ is a top vertex of $H$ and of $G$ . Let $X$ and $Y$ be chosen such that $X\cap Y=\emptyset$ . Let $F$ be an induced subgraph of a hypercube $Q$ with ground set $X\cup Y$ and vertex set $\{x\cup w\,:\, x\in V(H^{\prime})\}\cup \{u\cup y\,:\, y\in V(G^{\prime})\}$ . We see that $F$ contains a copy $F^{\prime}$ of $H\cup G$ and is contained in the edge layer $2k$ of $Q$ with the vertex $u\cup w$ playing a role of $v$ . Moreover, $V(F^{\prime})\cap \binom{X\cup Y}{2k}$ is an edge set of a $2k$ -partite hypergraph with parts $U_1,\ldots,U_k, W_1,\ldots, W_k$ .

Assume now that $v$ is a bottom vertex of $H$ and of $G$ . Let $X$ and $Y$ be chosen such that $X\cap Y=[k-1]$ and $u=w=[k-1]$ . Assume further that $i\in U_i$ , $i\in W_i$ , for each $i\in [k-1]$ . Let $F$ be an induced subgraph of a hypercube $Q$ with ground set $X\cup Y$ and vertex set $V(H^{\prime}) \cup V(G^{\prime})$ . We see that $F$ contains a copy $F^{\prime}$ of $H\cup G$ and is contained in the edge layer $k$ of $Q$ with vertex $v^{\prime}=[k-1]$ playing a role of $v$ . Moreover, $V(F^{\prime})\cap \binom{X\cup Y}{k}$ is an edge set of a $k$ -partite hypergraph with parts $U_1\cup W_1,\ldots,U_k\cup W_k$ .

Proof. Let $Z$ have a $k$ -partite representation in a hypercube $Q_n$ with ground set $[n]$ . We need the following notation. For any $x\in V(Q_n)$ , let $\textrm{Up}(x)$ be the up set of $x$ , that is, $\textrm{Up}(x) = \{ y \subseteq [n]\,:$ $x\subseteq y\}$ . Note that $\textrm{Up}(x)$ induces a graph isomorphic to $Q_m$ , for $m= n-|x|$ .

We need to consider four cases according to whether the odd vertices of $Z$ correspond to $k$ - or $(k-1)$ -element sets in the representation and whether we are embedding the odd vertices in $V_j$ or in $V_{j-1}$ . Note, however, that we can assume that the odd vertices correspond to the $k$ -element sets of the representation by selecting the vertices of other partite set of $Z$ as odd vertices instead.

So, from now on, we assume first that the odd vertices of $Z$ correspond to the $k$ -element sets of the representation.

We shall first a copy of $Z$ in $G$ with odd vertices in $V_{j}$ .

Let $U_x= U_{x,k}$ be the intersection of $G$ and the $k$ th edge layer of the hypercube $Q$ induced by $\textrm{Up}(x)$ , for each $x\in V_{j-k}$ . Note that $U_x$ is a subgraph of $L_j$ , the $j$ th layer of $Q_n$ . We call a vertex $y$ of $U_x$ a full vertex if it has degree $k$ in $U_x$ , the largest possible degree. Let $u_x$ be the number of full vertices in $U_x$ . We shall argue that there is a vertex $x \in V_{j-k}$ such that $u_x$ is large.

Let $t$ be the number of $k$ -edge stars in $G$ with the centre in $V_{j}$ . Then

\begin{equation*}t = \sum _{y\in V_{j}} \binom {d_G(y)}{k} \geq |V_{j}| \binom {\gamma ||L_j||/|V_{j}|}{k}.\end{equation*}

Each such a star corresponds to a full vertex in $U_x$ , for some $x$ . Thus, $\sum _{x\in V_{j-k}} u_x \geq t$ and there is a vertex $x$ such that

\begin{equation*}u_x \geq \frac {|V_{j}|}{|V_{j-k}|} \binom {\gamma ||L_j||/|V_{j}|}{k}.\end{equation*}

Since $n/2 -n^{2/3} \leq j\leq n/2 + n^{2/3}$ and $k$ is a fixed constant, we have that $||L_j||/|V_j| = \frac{n}{2}(1+o(1))$ and $|V_j|/|V_{j-k}| \geq c^{\prime}(k)$ , so

\begin{equation*}u_x \geq c(k)n^k,\end{equation*}

for positive constants $c(k)$ and $c^{\prime}(k)$ . Consider the full vertices in $U_x$ . They correspond to $u_x$ hyperedges in a $k$ -uniform hypergraph with the vertex set $[n]-x$ of size $n-j+k = \frac{n}{2} (1+o(1))$ . By a theorem of Erdős [Reference Erdős11], such a hypergraph contains any fixed $k$ -partite $k$ -uniform hypergraph, and thus in particular the one representing $Z$ . Therefore, $G$ contains a copy of $Z$ with odd vertices in $V_j$ .

Next, we shall find a copy of $Z$ in $G$ with odd vertices in $V_{j-1}$ . To do this, we repeat the above argument by considering the vertices $x$ in $V_{j+k-1}$ and their downsets. Alternatively, we see that $L_j$ corresponds to $L_{j^{\prime}}$ , a “symmetric” layer, where $j^{\prime}=n+1-j$ , $V_{j^{\prime}}$ corresponds to $V_{j-1}$ , and $V_{j^{\prime}-1}$ corresponds to $V_j$ . Thus, finding a copy of $Z$ in $L_{j^{\prime}}$ with odd vertices in $V_{j^{\prime}}$ corresponds to finding a copy of $Z$ in $L_j$ with odd vertices in $V_{j-1}$ . Since $j^{\prime}$ satisfies the same conditions as $j$ , that is, $n/2 -n^{2/3} \leq j^{\prime} \leq n/2 + n^{2/3}$ , we thus could use the first part of the Lemma to obtain the second one.

Proof. We shall prove the statement by induction on $\ell$ with the base case $\ell =1$ directly following from Lemma 4.

If $H$ has $\ell$ blocks, $\ell \geq 2$ , such that each has a $k$ -partite representation, let $H=H^{\prime}\cup H^{\prime\prime}$ , where $H^{\prime}$ and $H^{\prime\prime}$ share a single vertex $v^{\prime}$ , and $H^{\prime\prime}$ is a block of $H$ , that is, a leaf block. Let $F$ be a graph that is a union of $q\gt |V(H^{\prime})|$ copies $F_1, \ldots, F_q$ of $H^{\prime\prime}$ that pairwise share only a vertex corresponding to $v^{\prime}$ . Let $\gamma \gt 0$ and $n_0$ be sufficiently large (we shall specify how large later). Let $G\subseteq L_j$ , $||G||=\gamma ||L_j||$ .

Assume first that $v^{\prime}$ is an odd vertex.

The idea of the proof is as follows. We shall first find many copies of $H^{\prime}$ in $G$ for which the vertex corresponding to $v^{\prime}$ is in $V_j$ . Then we shall find a copy $F^*$ of $F$ and a copy $H^*$ of $H^{\prime}$ such that $V(F^*)\cap V(H^*) \cap V_j = \{v\}$ , where $v$ plays a role of $v^{\prime}$ . Finally, since $q$ is large enough, we shall claim that there is a copy $F_i^*$ of $F_i$ for some $i$ such that $V(F_i^*)\cap V(H^*) \cap V_{j-1} = \emptyset$ . This will imply that $F^*_i \cup H^*$ is a copy of $H$ . Next, we shall give the details of this argument:

First, we shall construct sets $V^{(1)}, V^{(2)},$ and $V^{(3)}$ , such that $V^{(3)}\subseteq V^{(2)} \subseteq V^{(1)}\subseteq V_j$ as follows:

Assume that $n_0\gt n_0(\gamma/4, H^{\prime})$ . Consider all copies of $H^{\prime}$ in $G$ with a vertex playing a role of $v^{\prime}$ in $V_j$ . Let $V^{(1)}\subseteq V_j$ be the set of all such vertices playing a role of $v^{\prime}$ in some copy of $H^{\prime}$ in $G$ . Note that $\big|\big|G\big[\big(V_j-V^{(1)}\big) \cup V_{j-1}\big]\big|\big| \lt \frac{1}{4} \gamma ||L_j||$ , otherwise by induction we can find another copy of $H^{\prime}$ with a vertex $w\in V_j-V^{(1)}$ playing a role of $v^{\prime}$ , contradicting the definition of $V^{(1)}$ . Note, that for each $v\in V^{(1)}$ , there could be several copies of $H^{\prime}$ with $v$ playing a role of $v^{\prime}$ . We choose one such copy arbitrarily and denote it $H^{\prime}_v$ . Let

\begin{equation*}V^{(2)} = \left \{y\in V^{(1)}\,:\, d_G(y)\geq \frac {\gamma }{2}\frac {||L_j||}{|V_j|} \right \}.\end{equation*}

The total number of edges of $G$ not incident to $V^{(2)}$ is at most $\frac{1}{4}\gamma ||L_j||+ \frac{1}{2}||G||= \frac{3}{4}\gamma ||L_j||$ . Thus, $\big|\big|G\big[V^{(2)} \cup V_{j-1}\big]\big|\big| \geq \frac{\gamma }{4}||L_j||$ . Since all vertices from $V_j$ in $L_j$ have the same degree,

\begin{equation*}|V^{(2)}| \geq \frac {\gamma }{4}|V_j|.\end{equation*}

Let for each $v\in V^{(2)}$ , $A_v\subseteq V_j$ and $B_v\subseteq V_{j-1}$ be sets of vertices such that $V(H^{\prime}_v)=A_v\cup B_v\cup \{v\}$ , $v\not \in A_v$ . Randomly colour each vertex in $V^{(2)}$ with red or blue independently with equal probability and colour the vertices in $V_j-V^{(2)}$ blue. We say that a vertex $v\in V^{(2)}$ is good if $v$ is red and each vertex in $A_v$ is blue. Then, the expected number of good $v$ ’s is at least $|V^{(2)}|2^{-t}$ , where $|A_v|=t-1$ . Thus, there is a set $V^{(3)}\subseteq V^{(2)}$ , corresponding to a set of good $v$ ’s, with $|V^{(3)}|\geq |V^{(2)}|2^{-t}$ , such that for each $v\in V^{(3)}$ , $A_v \cap V^{(3)} = \emptyset$ .

We see that

\begin{equation*}\big|\big|G\big[ V^{(3)} \cup V_{j-1}\big]\big|\big|\geq \frac {\gamma }{2}\frac {||L_j||}{|V_j|}|V^{(3)}| \geq \frac {\gamma }{2}\frac {||L_j||}{|V_j|} \frac {|V^{(2)}|}{2^{t}} \geq \gamma ^2 2^{-t-3} ||L_j||.\end{equation*}

Consider the graph $F$ defined above. By Lemma 3, $F$ has a partite representation. Then by Lemma 4, there is a copy $F^*$ of $F$ in $G\big[V^{(3)}\cup V_{j-1}\big]$ with a vertex $v\in V^{(3)}$ corresponding to $v^{\prime}$ . Here, we assume that $n_0\gt n_1\big(\gamma ^2 2^{-t-3}, F\big)$ . Let $F_i^*$ be a respective copy of $F_i$ , for each $i\in [q]$ . By construction of $V^{(3)}$ , there is a copy $H^*$ of $H^{\prime}$ in $G$ with all vertices except for $v$ not in $V^{(3)}$ . Let $B_v$ be the set of vertices of $H^*$ in $V_{j-1}$ . We see that $V(F^*) \cap V(H^*) \cap V_j = \{v\}$ . Since $q\gt |V(H^{\prime})|\gt |B_v|$ , there is $i \in [q]$ , such that $V(F^*_i)\cap B_v\cap V_{j-1} = \emptyset$ . Then $H^*\cup F_i^*$ is a copy of $H$ in $G$ .

The case when $v^{\prime}$ is not an odd vertex is treated similarly by first finding many copies of $H^{\prime}$ with a vertex corresponding to $v^{\prime}$ in $V_{j-1}$ .

Proof of Theorem 1. Let $G^{\prime}$ be a subgraph of $Q_n$ such that $||G^{\prime}|| = 2\gamma ||Q_n||$ , for some constant $\gamma \gt 0$ and sufficiently large $n$ . By a standard argument, see for example Lemma 1 [Reference Axenovich, Manske and Martin4],

\begin{equation*}\sum _{i:|i-n/2|\gt n^{2/3}} \binom {n}{i} = o(2^n).\end{equation*}

Since the degree of each vertex in $Q_n$ is $n$ , the total number of edges in $G^{\prime}$ , incident to vertices in $V_i$ ’s, for $i\lt n/2 - n^{2/3}$ or $ i \gt n/2 +n^{2/3}$ is $o(n2^n) = o(||Q_n||) = o(||G||)$ . Then there is $j\in \big\{n/2 - n^{2/3}, n/2 + n^{2/3}\big\}$ such that $L_j$ contains at least $\gamma ||L_i||$ edges of $G^{\prime}$ . Lemma 5 applied to $G= G^{\prime}\cap L_j$ concludes the proof.

Proof of Theorem 2. Let $H=H(q)$ be a union of two copies $\Theta _1$ and $\Theta _2$ of $\Theta (q)$ sharing exactly one vertex that is a main pole of $\Theta _2$ and a subdivision vertex of $\Theta _1$ .

First, we need to check that $H(q)$ is cubical. Marquardt [Reference Marquardt15] gave an explicit embedding of $H(3)$ in a hypercube. We show that $H(q)$ is cubical by constructing its nice colouring by taking nice colourings of $\Theta _1$ and $\Theta _2$ using disjoint sets of colours. These colourings were given in [Reference Havel and Morávek14] and we show one in Fig. 1.

Next, we shall show that $H(q)$ has no partite representation. The Hamming distance between two binary vectors $u$ and $v$ , denoted $d_H(u,v)$ is the number of positions where the vectors differ. In [Reference Axenovich, Martin and Winter2], it is shown that if $q\geq 3$ and $v, v^{\prime}$ are the main poles of a copy of $\Theta (q)$ embedded in a layer of a hypercube, then $d_H(v, v^{\prime})=2$ . Let the main poles of $\Theta _1$ and $\Theta _2$ be denoted $v_i, v^{\prime}_i$ , respectively, $i=1,2$ . Assume that $H$ has a $k$ -partite representation for some $k$ , that is, $V(H)\subseteq \binom{[n]}{k}\cup \binom{[n]}{k-1}$ , for some $n$ and $V(H)\cap \binom{[n]}{k}$ corresponds to the edges of a $k$ -partite hypergraph, denote it by $\mathcal{H}$ . We have that $d_H(v_i, v^{\prime}_i)=2$ , $i=1,2$ . Since a main pole of $\Theta _1$ and a main pole of $\Theta _2$ are adjacent, the main poles of $\Theta _1$ are in one vertex layer and the main poles of $\Theta _2$ are in another vertex layer, without loss of generality, $v_1, v^{\prime}_1 \in \binom{[n]}{k}$ and $v_2, v^{\prime}_2 \in \binom{[n]}{k-1}$ . Assume further that $v_1$ and $v^{\prime}_1$ equal to $10$ and $01$ in the first two positions of their binary representation and coincide on all other positions. Consider neighbours of $v_1$ in $\Theta _1$ . At least one of these neighbours, say $w$ differs from $v_1$ in a position that is not one of the first two ones, say in the third position. Then, restricted to the first three position $v_1$ is $101$ , $w$ is $100$ , and $v^{\prime}_1$ is $011$ . Thus, the remaining two vertices on the $v_1, v^{\prime}_1$ path, that contains $w$ , must be equal to $110$ and $010$ in these three position. All vertices of this leg are the same in positions $4, \ldots, n$ , say they are equal to $1$ on a set of positions $A\subseteq \{4, \ldots, n\}$ , $|A|=k-2$ . Thus, we have that $\mathcal{H}^{\prime}$ contains hyperedges $\{1, 2\}\cup A$ , $\{2,3\} \cup A$ , and $\{1, 3\} \cup A$ . If $\mathcal{H}$ were to be $k$ -partite, $1, 2, 3$ and each element of $A$ would belong to distinct parts, that is, to $k+1$ parts, a contradiction. This shows that $H(q)$ has no partite representation.

Finally, we shall show that $H(q)$ has zero Turán density in a hypercube. Using Theorem 1 it is sufficient to show that $\Theta (q)$ has partite representation. Note that $\Theta (q) = K_{2,q}(1)$ , a subdivision of $K_{2,q}$ . We claim that $\mathcal{H} = K_{2,q}$ gives a $2$ -partite representation of $\Theta (q)$ . Indeed, let $n= q+2$ and let $\mathcal{H}$ be a hypergraph on a vertex set $[n]$ with the edge set $\{ij\,:\, i\in [2], j\in [n]\setminus [2]\}$ . So, $\mathcal{H}$ is a bipartite graph, that is, $2$ -partite $2$ -uniform hypergraph. Let the edges of $\mathcal{H}$ correspond to subdivision vertices of $K_{2,q}(1)$ . Let vertices $v$ and $v^{\prime}$ correspond to the main poles of $\Theta (q)$ and vertices $1, \ldots, q$ correspond to other vertices. Then, the two parts of $\mathcal{H}$ are $\{v,v^{\prime}\}$ and $\{1, \ldots, q\}$ . This concludes the proof of Theorem 2.