Proof. Consider $\varepsilon \gt 0$ and $n$ , and let $n_0$ be the Ramsey number such that every $2$ -edge-coloured $n$ -uniform complete hypergraph with $n_0$ vertices contains a monochromatic complete hypergraph with $K=\max \{n^2-n+1,\lceil n/\varepsilon \rceil +2\}$ vertices. Fix an $N$ -partitioned hypergraph $H$ with $N\ge n_0$ , the set $I_0\subseteq \{1,\ldots ,N\}$ such that $|I_0|\ge n_0$ , and the subsets $W_{ijk}\subseteq V_{ik}$ , $i\lt j\lt k$ , $i,j,k\in I_0$ such that $|W_{ijk}|\ge \varepsilon |V_{ik}|$ .

We now construct an auxiliary $2$ -edge-coloured $n$ -uniform complete hypergraph with vertex set $I_0$ such that the edge formed by indices $i_1\lt \cdots \lt i_n$ , $i_1,\ldots ,i_n\in I_0$ , is coloured blue if the sets $W_{i_1,i_k,i_n}$ , $k=2,\ldots ,n-1$ , have a non-empty intersection, and it is coloured red otherwise, that is, the intersection of the sets $W_{i_1,i_k,i_n}$ , $k=2,\ldots ,n-1$ , is the empty set. By the choice of $n_0$ , there exists a $K$ -element set $J\subseteq I_0$ such that all edges formed by the elements of $J$ have the same colour. Let $i_1\lt \cdots \lt i_K$ be the elements of $J$ .

First suppose that the common colour of the edges formed by the elements of $J$ is red. By a simple averaging argument, the set $V_{i_1,i_K}$ contains a vertex that is contained in at least $\frac {\varepsilon (K-2)|V_{i_1,i_K}|}{|V_{i_1,i_K}|}=\varepsilon (K-2)\ge n$ sets $W_{i_1,i_k,i_K}$ , $k\in \{2,\ldots ,K-1\}$ ; let $x$ be any such vertex of $V_{i_1,i_K}$ . However, any $n-2$ indices $i_k$ , $k\in \{2,\ldots ,K-1\}$ , such that $x\in W_{i_1,i_k,i_K}$ together with $i_1$ and $i_K$ form a blue edge. Hence, the common colour of the edges formed by the elements of $J$ cannot be red.

We next show that the set $I=\{i_1,i_{n+1},i_{2n+1},\ldots ,i_{(n-1)n+1}\}$ has the property from the statement of the lemma. To do so, we need to find elements $\gamma _{i_a,i_b}$ for all $a,b\in \{1,n+1,\ldots ,(n-1)n+1\}$ such that $a\lt b$ . Fix such $i_a$ and $i_b$ . Since the edge of the auxiliary complete hypergraph formed by the $(b-a)/n+1\le n$ elements $i_a,i_{a+n},\ldots ,i_b$ and any $n-(b-a)/n-1$ elements among $a_{a+1},\ldots ,a_{a+n-1}$ is blue, the sets $W_{i_a,i_k,i_b}$ , $k=a+n,\ldots ,b-n$ , have a non-empty intersection, and we set $\gamma _{i_a,i_b}$ to be any element contained in their intersection.

Proof. Fix $\varepsilon \in (0,1)$ and $n$ , and let $K=\lfloor \varepsilon ^{-1}+1\rfloor ^3$ . Let $n_0$ be such that every $K$ -edge-coloured complete $3$ -uniform hypergraph with $n_0$ vertices contains an $n$ -vertex monochromatic complete hypergraph; such $n_0$ exists by Ramsey’s Theorem. Let $H$ be an $N$ -partitioned hypergraph with $N\ge n_0$ , and set

\begin{align*} s_{ij\to k} & = \frac {|\{v\in V_{ij}\mbox{ with }d_{ij\to k}(v)\ge \varepsilon \}|}{|V_{ij}|},\\[1pt] s_{jk\to i} & = \frac {|\{v\in V_{jk}\mbox{ with }d_{jk\to i}(v)\ge \varepsilon \}|}{|V_{jk}|},\mbox{ and}\\[1pt] s_{ik\to j} & = \frac {|\{v\in V_{ik}\mbox{ with }d_{ik\to j}(v)\ge \varepsilon \}|}{|V_{ik}|}. \end{align*}

We next construct an auxiliary $K$ -edge-coloured complete $3$ -uniform hypergraph $H'$ with $N$ vertices: the colour of the edge formed by the $i$ -th, $j$ -th and $k$ -th vertex of $H'$ is the triple $\left (\lfloor s_{ij\to k}/\varepsilon \rfloor ,\lfloor s_{jk\to i}/\varepsilon \rfloor ,\lfloor s_{ik\to j}/\varepsilon \rfloor \right )$ . By the choice of $N$ , there exist a subset $I\subseteq \{1,\ldots ,N\}$ with $|I|=n$ and a triple $(a,b,c)$ such that all edges formed by the $i$ -th, $j$ -th and $k$ -th vertex of $H'$ have the colour $(a,b,c)$ whenever $i,j,k\in I$ . Since the set $I$ and the reals $a\cdot \varepsilon$ , $b\cdot \varepsilon$ , and $c\cdot \varepsilon$ have the properties given in the statement of the lemma, the proof of the lemma is finished.

Proof. Fix a $\Phi _3$ -colourable hypergraph $H$ and let $n$ be the number of vertices of $H$ . By Theorem4, it is enough to show that for every $\delta \gt 0$ , there exists $N_0$ such that every $N_0$ -partitioned hypergraph with density at least $8/27+\delta$ embeds $H$ . Fix $\delta \in (0,1)$ . We next choose values $N_0,\ldots ,N_8$ such that $N_0\gg N_1\gg \cdots \gg N_7\gg N_8$ . To do so, set $\varepsilon =\delta /20$ , $N_8=n$ , and the values of $N_0,\ldots ,N_7$ as follows.

• • $N_7$ is the value of $n_0$ from Lemma 5 applied with $\varepsilon$ and $N_8$ .

• • $N_6$ is the value of $n_0$ from Lemma 7 applied with $\varepsilon$ and $N_7$ .

• • $N_5$ is the value of $n_0$ from Lemma 5 applied with $\varepsilon$ and $N_6$ .

• • $N_4$ is the value of $n_0$ from Lemma 6 applied with $\varepsilon$ and $N_5$ .

• • $N_3$ is the value of $n_0$ from Lemma 7 applied with $\varepsilon$ and $N_4$ .

• • $N_2$ is the value of $n_0$ from Lemma 6 applied with $\varepsilon$ and $N_3$ .

• • $N_1$ is the value of $n_0$ from Lemma 8 applied with $\varepsilon$ and $N_2$ .

• • $N_0$ is the value of $n_0$ from Lemma 9 applied with $2\varepsilon$ and $N_1$ .

For this value of $N_0$ , we will show that every $N_0$ -partitioned hypergraph with density at least $8/27+\delta$ embeds $H$ .

Fix an $N_0$ -partitioned hypergraph $H_0$ with density at least $8/27+\delta$ . By Lemma 9, there exist $I_1\subseteq \{1,\ldots ,N_0\}$ with $|I_1|\ge N_1$ and reals $a$ , $b$ , and $c$ such that

\begin{align*} a & \le \frac {|\{v\in V_{ij}\mbox{ with }d_{ij\to k}(v)\ge 2\varepsilon \}|}{|V_{ij}|} \lt a+2\varepsilon ,\\[2pt] b & \le \frac {|\{v\in V_{jk}\mbox{ with }d_{jk\to i}(v)\ge 2\varepsilon \}|}{|V_{jk}|} \lt b+2\varepsilon ,\mbox{ and}\\[2pt] c & \le \frac {|\{v\in V_{ik}\mbox{ with }d_{ik\to j}(v)\ge 2\varepsilon \}|}{|V_{ik}|} \lt c+2\varepsilon , \end{align*}

for all $i,j,k\in I_1$ and $i\lt j\lt k$ .

We next show that $a+b+c$ is at least $2+3\varepsilon$ , which we later show to imply that for every choice of $k\lt i\lt k'\lt j\lt k''$ from $I_1$ the set $V_{ij}$ contains linearly many vertices $v$ such that $d_{ij\to k''}(v)\ge 2\varepsilon$ , $d_{ij\to k}(v)\ge 2\varepsilon$ , and $d_{ij\to k'}(v)\ge 2\varepsilon$ . Consider any of the triads, say the $(1,2,3)$ -triad. Let $A$ be the set of all vertices $v\in V_{12}$ with $d_{12\to 3}(v)\ge 2\varepsilon$ , $B$ the set of all vertices $v\in V_{23}$ with $d_{23\to 1}(v)\ge 2\varepsilon$ , and $C$ the set of all vertices $v\in V_{13}$ with $d_{13\to 2}(v)\ge 2\varepsilon$ . Since each vertex of $V_{12}\setminus A$ is in at most $2\varepsilon \cdot |V_{23}|\cdot |V_{13}|$ edges of the $(1,2,3)$ -triad, each vertex of $V_{23}\setminus B$ is in at most $2\varepsilon \cdot |V_{12}|\cdot |V_{13}|$ edges of the $(1,2,3)$ -triad, and each vertex of $V_{13}\setminus C$ is in at most $2\varepsilon \cdot |V_{12}|\cdot |V_{23}|$ edges of the $(1,2,3)$ -triad, the $(1,2,3)$ -triad has at most

\begin{equation*}|A|\cdot |B|\cdot |C|+6\varepsilon \cdot |V_{12}|\cdot |V_{13}|\cdot |V_{23}| \le \left (abc+12\varepsilon \right )\cdot |V_{12}|\cdot |V_{13}|\cdot |V_{23}| \end{equation*}

edges. Since the density of the $N_0$ -partitioned hypergraph $H_0$ is at least $8/27+\delta$ , if follows that $abc+12\varepsilon$ is at least $8/27+\delta =8/27+20\varepsilon$ and so $abc\ge 8/27+8\varepsilon$ . Using the Inequality of Arithmetic and Geometric Means, we obtain that

\begin{equation*}a+b+c \ge 3\sqrt [3]{abc} \ge 3\sqrt [3]{8/27+8\varepsilon } \ge 3(2/3+\varepsilon )=2+3\varepsilon .\end{equation*}

Our next step is to apply Lemma 8 to identify vertices $\omega _{ij}$ such that $d_{ij\to k''}(\omega _{ij})\ge 2\varepsilon$ , $d_{ij\to k}(\omega _{ij})\ge 2\varepsilon$ , and $d_{ij\to k'}(\omega _{ij})\ge 2\varepsilon$ for every choice of $k\lt i\lt k'\lt j\lt k''$ from a suitable index set $I_2$ ; the vertices $\omega _{ij}$ will correspond to the colour $\omega$ from the palette $\Phi _3$ . To apply the lemma, we define sets $W_{ij}^{kk'k''}\subseteq V_{ij}$ for $i,j,k,k',k''\in I_1$ such that $k\lt i\lt k'\lt j\lt k''$ . Consider such $k\lt i\lt k'\lt j\lt k''$ . Let $A$ be the set of all $v\in V_{ij}$ with $d_{ij\to k''}(v)\ge 2\varepsilon$ , $B$ the set of all vertices $v\in V_{ij}$ with $d_{ij\to k}(v)\ge 2\varepsilon$ , and $C$ the set of all vertices $v\in V_{ij}$ with $d_{ij\to k'}(v)\ge 2\varepsilon$ . Let $m_d$ , $d\in \{0,1,2,3\}$ , be the number of vertices of $V_{ij}$ contained in exactly $d$ sets among $A$ , $B$ , and $C$ . Observe that

\begin{equation*}(a+b+c)|V_{ij}|\le |A|+|B|+|C|=m_1+2m_2+3m_3\le 2(m_1+m_2)+3m_3\le 2|V_{ij}|+3m_3,\end{equation*}

which implies that there are at least $\frac {a+b+c-2}{3}|V_{ij}|\ge \varepsilon |V_{ij}|$ vertices of $V_{ij}$ contained in $A\cap B\cap C$ . We set $W_{ij}^{kk'k''}=A\cap B\cap C$ . Lemma 8 implies that there exist $I_2\subseteq I_1$ with $|I_2|\ge N_2$ and $\omega _{ij}\in V_{ij}$ , $i,j\in I_2$ , and $i\lt j$ , such that $\omega _{ij}\in W_{ij}^{kk'k''}$ for all $i,j,k,k',k''\in I_2$ such that $k\lt i\lt k'\lt j\lt k''$ . In particular, it holds that $d_{ij\to k}(\omega _{ij})\ge 2\varepsilon$ for all $i,j\in I_2$ , $i\lt j$ , and $k\in I_2\setminus \{i,j\}$ (regardless whether $k\lt i\lt j$ , $i\lt k\lt j$ , or $i\lt j\lt k$ ).

Our next goal is to identify vertices that will correspond to the colours $\alpha ^1$ and $\beta ^1$ of the palette $\Phi _3$ . To identify those corresponding to the colour $\alpha ^1$ , we will apply Lemma 6. Consider $i,j,k\in I_2$ such that $i\lt j\lt k$ and let $W_{ijk}$ be the set of all vertices $v\in V_{ij}$ contained together with $\omega _{ik}$ in at least $\varepsilon |V_{jk}|$ edges of the $(i,j,k)$ -triad. Since the number of edges containing $\omega _{ik}$ in the triad $(i,j,k)$ -triad is at most

\begin{equation*}|W_{ijk}|\cdot |V_{jk}|+\varepsilon |V_{ij}\setminus W_{ijk}|\cdot |V_{jk}|\le |W_{ijk}|\cdot |V_{jk}|+\varepsilon |V_{ij}|\cdot |V_{jk}|\end{equation*}

and $d_{ik\to j}(\omega _{ik})\ge 2\varepsilon$ , we obtain that $W_{ijk}$ contains at least $\varepsilon |V_{ij}|$ vertices. Lemma 6 yields that there exist $I_3\subseteq I_2$ with $|I_3|\ge N_3$ and $\alpha ^1_{ij}\in V_{ij}$ , $i,j\in I_3$ , and $i\lt j$ , such that $\alpha ^1_{ij}$ and $\omega _{ik}$ are contained together in at least $\varepsilon |V_{jk}|$ edges of the $(i,j,k)$ -triad for all $i,j,k\in I_3$ such that $i\lt j\lt k$ .

We next apply Lemma 7 to identify the vertices corresponding to the colour $\beta ^1$ . For $i,j,k\in I_3$ such that $i\lt j\lt k$ , we set $W_{ijk}$ to be the set of all vertices $v\in V_{jk}$ that form together with $\alpha ^1_{ij}$ and $\omega _{ik}$ an edge of the $(i,j,k)$ -triad; note that $W_{ijk}$ contains at least $\varepsilon |V_{jk}|$ vertices. Hence, Lemma 7 implies that there exist $I_4\subseteq I_3$ with $|I_4|\ge N_4$ and $\beta ^1_{jk}\in V_{jk}$ , $j,k\in I_4$ , and $j\lt k$ , such that $\alpha ^1_{ij}$ , $\beta ^1_{jk}$ , and $\omega _{ik}$ form an edge of the $(i,j,k)$ -triad for all $i,j,k\in I_4$ such that $i\lt j\lt k$ . In particular, the vertices $\alpha ^1_{ij}$ , $\beta ^1_{jk}$ , and $\omega _{ik}$ of the $(i,j,k)$ -triad indeed correspond to the colours $\alpha ^1$ , $\beta ^1$ , and $\omega$ of the palette $\Phi _3$ .

In the completely analogous way, we apply Lemmas 6 and 5 (in this order) to obtain $I_6\subseteq I_4$ with $|I_6|\ge N_6$ , $\alpha ^2_{ij}$ , $i,j\in I_6$ and $i\lt j$ , and $\gamma ^2_{ik}$ , $i,k\in I_6$ , and $i\lt k$ , such that $\alpha ^2_{ij}$ , $\omega _{jk}$ , and $\gamma ^2_{ik}$ form an edge of the $(i,j,k)$ -triad for all $i,j,k\in I_6$ such that $i\lt j\lt k$ , and so the vertices $\alpha ^2_{ij}$ and $\beta ^2_{ij}$ correspond to the colours $\alpha ^2$ and $\beta ^2$ of the palette $\Phi _3$ . Then, we apply Lemmas 7 and 5 to obtain $I_8\subseteq I_6$ with $|I_8|\ge N_8$ , $\beta ^3_{jk}$ , $j,k\in I_6$ , and $j\lt k$ , and $\gamma ^3_{ik}$ , $i,k\in I_6$ and $i\lt k$ , such that $\omega _{ij}$ , $\beta ^3_{jk}$ , and $\gamma ^3_{ik}$ form an edge of the $(i,j,k)$ -triad for all $i,j,k\in I_8$ such that $i\lt j\lt k$ ; again, the vertices $\alpha ^3_{ij}$ and $\beta ^3_{ij}$ correspond to the colours $\alpha ^3$ and $\beta ^3$ of the palette $\Phi _3$ .

We now argue that the $N_0$ -partitioned hypergraph $H_0$ embeds the hypergraph $H$ . By the assumption of the theorem, the hypergraph $H$ is $\Phi _3$ -colourable. Let $v_1,\ldots ,v_n$ be the vertices of $H$ listed in the order such that there exists a choice of $c_{ij}\in \{\omega ,\alpha ^1,\beta ^1,\alpha ^2,\gamma ^2,\beta ^3,\gamma ^3\}$ , $i,j\in \{1,\ldots ,n\}$ such that if vertices $v_i$ , $v_j$ , and $v_k$ , $1\le i\lt j\lt k\le n$ , form an edge of $H$ , then $(c_{ij},c_{jk},c_{ik})\in \Phi _3$ . Let $a_i$ , $i\in \{1,\ldots ,n\}$ , be the $i$ -th smallest index contained in $I_8$ (note that $I_8$ has at least $N_8=n$ elements). For $1\le i\lt j\le n$ , set $w_{ij}$ among the vertices of $V_{ij}$ as follows:

\begin{align*} w_{ij}=\begin{cases} \omega _{a_ia_j} & \mbox{if $c_{ij}=\omega $,}\\[2pt] \alpha ^1_{a_ia_j} & \mbox{if $c_{ij}=\alpha ^1$,}\\[2pt] \beta ^1_{a_ia_j} & \mbox{if $c_{ij}=\beta ^1$,}\\[2pt] \alpha ^2_{a_ia_j} & \mbox{if $c_{ij}=\alpha ^2$,}\\[2pt] \gamma ^2_{a_ia_j} & \mbox{if $c_{ij}=\gamma ^2$,}\\[2pt] \beta ^3_{a_ia_j} & \mbox{if $c_{ij}=\beta ^3$, and}\\[2pt] \gamma ^3_{a_ia_j} & \mbox{if $c_{ij}=\gamma ^3$.} \end{cases} \end{align*}

The choice of the vertices $\omega _{ij}$ , $\alpha ^1_{ij}$ , $\beta ^1_{ij}$ , $\alpha ^2_{ij}$ , $\gamma ^2_{ij}$ , $\beta ^3_{ij}$ , and $\gamma ^3_{ij}$ , $i,j\in I_8$ and $i\lt j$ , yields that if vertices $v_i$ , $v_j$ , and $v_k$ , $1\le i\lt j\lt k\le n$ , form an edge of $H$ , then the vertices $w_{ij}$ , $w_{jk}$ , and $w_{ik}$ form an edge of the $(a_i,a_j,a_k)$ -triad. We conclude that the $N_0$ -partitioned hypergraph $H_0$ embeds $H$ and the proof of theorem is finished.

Proof of Proposition 11. Fix an $n$ -vertex $m$ -edge $5$ -uniform linear hypergraph $H_0$ such that $n!\lt (10/9)^m$ . We construct the $n$ -vertex $(3m)$ -edge $3$ -uniform hypergraph $H$ in a random way. The vertex set of $H$ is the same as that of $H_0$ . In each edge $e$ of $H_0$ choose two vertices $v$ and $v'$ randomly (independently of the other edges) and include to $H$ as edges the three triples containing $v$ , $v'$ and one of the three remaining vertices of $e$ ; see Fig. 1 for an illustration of the construction. We now verify that the hypergraph $H$ is $\Phi _3$ -colourable and that it is not $\Phi _8$ -colourable with positive probability.

We start by verifying that the hypergraph $H$ is $\Phi _3$ -colourable. Fix any order $v_1,\ldots ,v_n$ of the vertices of $H$ (and so of $H_0$ ). If a pair of vertices $v_i$ and $v_j$ of $H_0$ is not contained in an edge of $H_0$ , choose $c_{ij}$ arbitrarily. Otherwise, consider any edge $e$ of $H_0$ , and let $v_i$ and $v_j$ , $i\lt j$ , be the two vertices chosen in the construction of $H$ . Set $c_{ij}=\omega$ . For each of the three remaining vertices of $e$ , say $v_k$ , set $c_{ik}=\alpha ^1$ and $c_{kj}=\beta ^1$ if $i\lt k\lt j$ , set $c_{ki}=\alpha ^2$ and $c_{kj}=\gamma ^2$ if $k\lt i$ , and set $c_{ik}=\beta ^3$ and $c_{jk}=\gamma ^3$ if $k\gt j$ . Since the hypergraph $H$ is linear, the values of $c_{ij}$ , $1\le i\lt j\le n$ , are well-defined. It is straightforward to verify that if $\{v_i,v_j,v_k\}$ , $1\le i\lt j\lt k\le n$ , is an edge of $H$ , then $(c_{ij},c_{jk},c_{ik})\in \Phi _3$ . Hence, the hypergraph $H$ is $\Phi _3$ -colourable.

We next show that the hypergraph $H$ is not $\Phi _8$ -colourable with positive probability. Fix any order $v_1,\ldots ,v_n$ of the vertices of $H$ . Since the $5$ -uniform hypergraph $H_0$ is linear, the hypergraph $H$ is not $\Phi _3$ -colourable with respect to the fixed order $v_1,\ldots ,v_n$ if and only if the $5$ -uniform hypergraph $H_0$ has an edge $\{v_a,v_b,v_c,v_d,v_e\}$ , $1\le a\lt b\lt c\lt d\lt e\le n$ , such that the two chosen vertices in the construction of $H$ are the vertices $v_b$ and $v_d$ . Indeed, if $H_0$ has such an edge, then there is no choice of a colour $c_{bd}$ such that $(c_{ab},c_{bd},c_{ad})\in \Phi _8$ , $(c_{bc},c_{cd},c_{bd})\in \Phi _8$ , and $(c_{bd},c_{de},c_{be})\in \Phi _8$ . And if $H_0$ has no such edge, we can choose the colours of the pairs of the vertices $v_a,v_b,v_c,v_d,v_e$ within each edge of $H_0$ and this choice does not affect the choices within other edges of $H_0$ as the hypergraph $H_0$ is linear. Hence, for any fixed order of the vertices $v_1,\ldots ,v_n$ , the probability that there exists an assignment of colours to the pairs of vertices such that $(c_{ij},c_{jk},c_{ik})\in \Phi _8$ for every edge $\{v_i,v_j,v_k\}$ , $1\le i\lt j\lt k\le n$ , of $H_0$ is equal to $(9/10)^m$ . The union bound implies that the probability that there exists a vertex ordering that admits such an assignment of colours to the pairs of vertices is at most $n!\cdot (9/10)^m\lt 1$ , that is, the hypergraph $H$ is not $\Phi _8$ -colourable with probability at least $1-n!\cdot (9/10)^m\gt 0$ .