3.1. A Ramsey problem

Note that $D$ may contain bidirectional edges, which can be viewed as directed cycles of length two; see Fig. 1 for an example. To clarify, the induced colouring inherits the vertex ordering as well as the colours of the edges.

Figure 1. An example of dependency digraph.

The main result of this section is the following embedding result which relates the problem of upper bounding $g(k)$ to the Ramsey problem for $f(k)$ .

Theorem 3.2. For any integer $k \geqslant 1,$ and real $\eta \gt 0,$ there exists $c \gt 0$ such that the following holds. Let $G = G_\lt$ be an ordered graph with no induced monotone $P_{2k}$ . If $\overline {G}$ contains at least $\eta n^{f(k)}$ copies of $K_{f(k)}$ , then $\overline {G}$ contains a ${K_2}[{{cn} \over {\log n}}]$ . In other words,

\begin{align*}g(k)\leqslant f(k).\end{align*}

Let us first derive that $g(3)=f(3)=3$ and $g(4)=f(4)=5$ , which imply Theorems 1.2 and 1.3. Recall that the optimal construction for $\beta (5)=1/2$ infers that $g(3)\gt 2$ and the construction in Theorem 1.4 shows that $g(4)\gt 4$ . Thus, by Theorem 3.2, it suffices to prove $f(3)\leqslant 3$ and $f(4)\leqslant 5$ .

We now prove an upper bound for the general case, which together with Theorem 3.2, implies the upper bound for $g(k)$ in Theorem 1.4.

Hence, ${{{k^2} - k + 2} \over 2} = \sum\limits_{t \in [k - 1]} | {S_t}| \leqslant \sum\limits_{t \in [k - 1]} {(k - t)} = {{{k^2} - k} \over 2},$ a contradiction.

3.2. Proof of Theorem 3.2

By taking sufficiently small $c$ , we may assume that $n$ is sufficiently large in terms of $k$ and $\eta .$ We define additional constants $c, \alpha , \beta , \gamma , \gamma _1, \gamma _2, \varepsilon , \varepsilon _1$ , such that

\begin{align*}0 \lt 1/n \ll c \ll \alpha \ll \gamma =\gamma _{1}\gamma _{2} \lt \gamma _1 \ll \varepsilon _1 \ll \gamma _2 \ll \varepsilon \ll \beta \ll \eta \lt 1/k. \end{align*}

Suppose, for the sake of contradiction, that $\overline {G}$ does not contain a copy of ${K_2}[{{cn} \over {\log n}}]$ .

By Lemma 2.3, there exist sets $A_1, \dots , A_{f(k)} \subseteq V(G)$ such that $A_1 \lt A_2 \lt \dots \lt A_{f(k)},$ for all $i \in [f(k)], |A_i| \geqslant \gamma _1 n$ and for all $1 \leqslant i \lt j \leqslant f(k),$ $(A_i, A_j)$ is $\varepsilon _1$ -regular with density at least ${\eta \over 2}$ in $\overline {G}.$

We define an edge-colouring $\chi _0$ of the complete graph on the ordered vertex set $\{1,2,\ldots , f(k)\}$ as follows. For $1 \leqslant i \lt j \leqslant f(k)$ , let

\begin{align*}\chi _0(ij) = \begin{cases} \mathrm{blue}, &\text{if } d(A_i, A_j) \geqslant \beta ^2\\ \mathrm{red}, &\text{otherwise}. \end{cases} \end{align*}

By definition of $f(k),$ there exist indices $a_1 \lt \dots \lt a_k$ such that $\chi _0[\{a_1, \dots , a_k\}]$ is admissible. For $i \in [k],$ let $B_i = A_{a_i}.$ Moreover, define a colouring $\chi \colon \binom{[k]}{2} \rightarrow \{ \mathrm{red}, \mathrm{blue} \}$ by setting $\chi (ij) = \chi _0(a_ia_j)$ and note that $\chi$ is admissible. We now show that we can either construct an induced monotone $P_{2k}$ in $G$ by placing two consecutive vertices in each set $B_i$ or there is a biclique of size ${{cn} \over {\log n}}$ in $\overline {G},$ thus obtaining a contradiction.

Recall that $|B_i| \geqslant \gamma _1 n \geqslant \alpha n,$ so since $G$ is $(\alpha , \beta )$ -dense, we have $d(G[B_i]) \geqslant \beta .$ Applying Lemma 2.3 to $G[B_i]$ for each $i \in [k],$ we obtain two sets $C_{2i-1}, C_{2i} \subseteq B_i$ of size at least $\gamma _2 |B_i| \geqslant \gamma _2 \gamma _1 n=\gamma n$ with $C_{2i-1} \lt C_{2i}$ and such that $(C_{2i-1}, C_{2i})$ is $\varepsilon$ -regular in $G$ with density at least ${\beta \over 4}$ . To summarise, we have that $C_1 \lt C_2 \lt \dots \lt C_{2k},$ $|C_i| \geqslant \gamma _2 |B_i| \geqslant \gamma n,$ for all $i \in [2k].$ Moreover, using that ${{{\varepsilon _1}} \over {{\gamma _2}}}\lt \varepsilon $ , by Lemma 2.2, the pairs $(C_i, C_j)$ are $\varepsilon$ -regular. Finally, the following holds for the densities between different pairs. Let $1 \leqslant i \lt j \leqslant 2k,$ then

1. a) If $\lfloor {{i \over 2}} \rfloor = \lfloor {{j \over 2}} \rfloor $ , then $d({C_i},{C_j})\gt {\beta \over 4}$ .

2. b) If $\lfloor {{i \over 2}} \rfloor \ne \lfloor {{j \over 2}} \rfloor $ , then $d({C_i},{C_j})\lt 1 - {\eta \over 4}$ .

3. c) If $\chi (\lfloor {{i \over 2}} \rfloor ,\lfloor {{j \over 2}} \rfloor ) = {\rm{red}}$ , then $d(C_i, C_j) \lt 2 \beta ^2$ .

4. d) If $\chi (\lfloor {{i \over 2}} \rfloor ,\lfloor {{j \over 2}} \rfloor ) = {\rm{blue}}$ , then $d({C_i},{C_j})\gt {{{\beta ^2}} \over 2}$ .

Let $D = D(\chi )$ be the dependency digraph of $\chi$ as defined in Definition 3.1. By assumption, $D$ is acyclic. Consider a fixed topological ordering of $D$ and for $i \in [k]$ , let $\pi (i)$ denote the position of vertex $i$ in this topological ordering. In other words, we have for every edge $ij \in E(D)$ , $\pi (i) \lt \pi (j)$ .

Let $(i_1, i_1+1), (i_2, i_2+1), \dots , (i_{k-1}, i_{k-1}+1)$ be an ordering of the edges $(1, 2), \dots , (k-1, k)$ according to the earliest appearance in $\pi$ of one of their endpoints. In other words for $j \lt j',$ we have that $\min \{\pi (i_j), \pi (i_j+1)\} \leqslant \min \{\pi (i_{j'}), \pi (i_{j'}+1)\}$ . In particular, when processing $(i_t,i_t+1)$ , any $j$ with $\chi ( {\lfloor {{i \over 2}} \rfloor ,\lfloor {{j \over 2}} \rfloor } ) = \chi ( {\lfloor {{{i + 1} \over 2}} \rfloor ,\lfloor {{j \over 2}} \rfloor }) = {\rm{blue}}$ must satisfy $j\in I_t$ by the definition of $D$ and the choice of the order.

We shall construct a monotone induced path $x_1, \dots , x_{2k}$ with $x_j \in C_j$ for all $j \in [2k].$ We first embed all but $x_1$ and $x_{2k}$ in $k-1$ steps. At step $t \in [k-1],$ we shall define the vertices $x_{2i_t}, x_{2i_t+1}.$ Finally, we embed $x_1$ and $x_{2k}.$

Let us now formally describe the embedding procedure. For each $t\in [k]$ , just before executing step $t$ we maintain the following invariants:

1. (I1) The set of already embedded indices is

   \begin{align*}I_t \;=\; \bigcup _{j=1}^{t-1} \{\,2i_j,\; 2i_j+1\,\}\subseteq [2k]. \end{align*}

2. (I2) For every $i\in [2k]{\setminus} I_t$ there is a candidate set $C_i^t\subseteq C_i$ with the uniform lower bound

   \begin{align*}|C_i^t|\;\geqslant \; \beta ^{8(t-1)}\,|C_i|. \end{align*}

3. (I3) For any distinct $i,j\in [2k]{\setminus} I_t$ , the pair $(C_i^t,C_j^t)$ is regular (with parameter at most $\varepsilon$ ; and by Lemma 2.2 and the parameter hierarchy, whenever invoked below we may in fact use $\varepsilon ^{0.9}$ ). We also repeatedly use the density bounds a)–d) for the base pairs $(C_i,C_j)$ .

Formally, we initialise with $C_i^1=C_i$ for all $i\in [2k]$ . After $t-1$ steps, for each $i\in [2k]$ we define

\begin{align*}C_i^t \;=\; \Bigl ( C_i \cap \bigcap _{j \in I_t \cap \{i-1,\, i+1\}} N(x_j) \Bigr ) {\setminus} \bigcup _{j \in I_t {\setminus} \{i-1,\, i+1\}} N(x_j). \end{align*}

Here we require adjacency only to those of $x_{i-1},x_{i+1}$ that have already been embedded (i.e., $j\in I_t\cap \{i-1,i+1\}$ ), and we delete vertices adjacent to any other already embedded $x_j$ (i.e., $j\in I_t{\setminus} \{i-1,i+1\}$ ).

Given that we have run the procedure for $t-1 \leqslant k-2$ steps we show we can run step $t.$ Let $i = 2i_t$ and recall that we aim to embed vertices $x_i$ and $x_{i+1}.$ By assumption, we have sets $C^t_i$ and $C^t_{i+1}$ of size at least $\beta ^{8(t-1)} |C_i|$ into which we can embed the vertices $x_i$ and $x_{i+1}$ , respectively. Note that

\begin{align*}{{|C_i^t|} \over {|{B_i}|}} = {{|C_i^t|} \over {|{C_i}|}} \cdot {{|{C_i}|} \over {|{B_i}|}}{\beta ^{8(t - 1)}}{\gamma _2}\gt \varepsilon _1^{0.1},\end{align*}

which implies that for all $i, j \in [2k] {\setminus} I_t, i \neq j,$ the pair $(C^t_i, C^t_j)$ is $\varepsilon ^{0.9}$ -regular.

By the $\varepsilon ^{0.9}$ -regularity of $(C_i^t,C_j^t)$ , all but at most $\varepsilon ^{0.9}|C_i^t|$ vertices $v\in C_i^t$ satisfy

\begin{align*}\bigl |d\big(v,C_j^t\big)-d\big(C_i^t,C_j^t\big)\bigr |\lt \varepsilon ^{0.9}. \end{align*}

By Lemma 2.2 applied to $(C_i,C_j)$ with subsets $(C_i^t,C_j^t)$ we also have

\begin{align*}\bigl |d\big(C_i^t,C_j^t\big)-d\big(C_i,C_j\big)\bigr |\lt \varepsilon ^{0.9}. \end{align*}

Hence, for all but at most $\varepsilon ^{0.9}|C_i^t|$ vertices $v\in C_i^t$ ,

\begin{align*}\bigl |d\big(v,C_j^t\big)-d\big(C_i,C_j\big)\bigr |\lt 2\varepsilon ^{0.9}\leqslant \sqrt {\varepsilon }, \end{align*}

where the last inequality follows from the parameter hierarchy. Discarding the exceptional vertices for each of the fewer than $2k$ choices of $j$ and intersecting over $j$ , we obtain a set

\begin{align*}C_i'\subseteq C_i^t,\qquad |C_i'|\geqslant (1-2k\varepsilon ^{0.9})|C_i^t|\geqslant \tfrac 12|C_i^t|, \end{align*}

such that for every $v\in C_i'$ and $j\in [2k]{\setminus} (I_t\cup \{i,i+1\})$ ,

\begin{align*}\bigl |d\big(v,C_j^t\big)-d(C_i,C_j)\bigr |\lt \sqrt {\varepsilon }. (\clubsuit )\end{align*}

Now we consider two cases.

Case 1. $\chi (i_t, i_t+1) = \mathrm{blue}$ .

Note that $\lfloor {{i \over 2}} \rfloor = {i_t}$ and $\lfloor {{{i + 1} \over 2}} \rfloor = {i_t} + 1$ . By condition d), we have $d({C_i},{C_{i + 1}})\gt {{{\beta ^2}} \over 2}$ and thus by $\varepsilon$ -regularity and choices of parameters, $d({C'_i},{C'_{i + 1}})\gt {{{\beta ^2}} \over 2} - \sqrt \varepsilon \gt {\beta ^3}.$ In particular, there is a vertex $u \in C'_i$ with $|N(u) \cap {C'_{i + 1}}| \geqslant {{{\beta ^3}} \over 2} \cdot |{C'_{i + 1}}|$ . Pick such a vertex $u.$ For $j \in [2k] {\setminus} (I_t \cup \{i-1, i, i+1\})$ , let $C'_j = C^t_j {\setminus} N(u).$ Furthermore, define $C'_{i-1} = C^t_{i-1} \cap N_G(u)$ if $i-1 \not \in I_t$ .

Recall that $d({C_i},{C_j})\lt 1 - {\eta \over 4}$ for $j \in [2k] {\setminus} (I_t \cup \{i-1, i, i+1\})$ by condition b) and $d({C_{i - 1}},{C_i})\gt {\beta \over 4}$ by condition a). Therefore, for all $j \in [2k] {\setminus} (I_t \cup \{i, i+1\}),$ combining with $(\clubsuit )$ we have

\begin{align*}|{C'_j}| \geqslant {{{\beta ^3}} \over {100}} \cdot |C_j^t| \geqslant {\beta ^{8(t - 1) + 4}} \cdot \gamma n.\end{align*}

By $\varepsilon$ -regularity, there is a vertex $v \in N(u) \cap C'_{i+1}$ such that for all $j \in [2k] {\setminus} (I_t \cup \{i, i+1\}),$ we have

\begin{align*}\left |d\left(v,C'_j\right) - d(C_{i+1}, C_j)\right | \lt \sqrt {\varepsilon }.\end{align*}

Pick such a vertex $v$ and let $x_i = u$ and $x_{i+1} = v.$ Recalling that $d({C_{i + 1}},{C_j})\lt 1 - {\eta \over 4}$ for $j \neq i+2$ and $d({C_{i + 1}},{C_{i + 2}})\gt {\beta \over 4}$ , it follows that $|C^{t+1}_j| \geqslant \beta ^{8t} \cdot |C^t_j|$ for all $j \in [2k] {\setminus} I_{t+1}$ as needed.

Case 2. $\chi (i_t, i_t+1) = \mathrm{red}$ .

Since $|{C'_i}|,|{C'_{i + 1}}| \geqslant {1 \over 2} \cdot {\beta ^{8(t - 1)}}\gamma n\gt {n \over {\log n}}$ and $\overline {G}$ is ${K_2}[{{cn} \over {\log n}}]$ -free, there exists an edge $uv \in E(G)$ with $u \in C'_i, v \in C'_{i+1}.$ Set $x_i = u, x_{i+1} = v.$ We will show that for all $j \in [2k] {\setminus} I_{t+1},$ $|C^{t+1}_j| \geqslant \beta ^8 |C^t_j|$ , which is sufficient by our assumption.

Consider first $j = i-1.$ If $j \not \in [2k] {\setminus} I_t,$ there is nothing to prove. Otherwise, ${d_G}({C_j},{C_i})\gt {\beta \over 4}$ by condition a) and $d(C_j, C_{i+1}) \lt 2 \beta ^2$ by condition c). Hence, by the definitions of the sets $C'_i, C'_{i+1},$ we have

\begin{align*}|C_j^{t + 1}| = |C_j^t \cap N(u) {\setminus} N(v)| \geqslant |C_j^t| \cdot \left({\beta \over 4} - \sqrt \varepsilon - 2{\beta ^2} - \sqrt \varepsilon \right) \geqslant {\beta ^8}|C_t^j|.\end{align*}

Completely analogously, if $i+2 \in [2k] {\setminus} I_{t},$ we have $|C^{t+1}_{i+2}| \geqslant \beta ^8 |C^t_{i+2}|.$

Now, consider $j \in [2k] {\setminus} (I_{t} \cup \{i-1, i+2\})$ . If $\chi (\lfloor {{i \over 2}} \rfloor ,\lfloor {{j \over 2}} \rfloor ) = {\rm{red}}$ or $\chi (\lfloor {{{i + 1} \over 2}} \rfloor ,\lfloor {{j \over 2}} \rfloor ) = {\rm{red}},$ then $d({C_i},{C_j}) + d({C_{i + 1}},{C_j})\lt 2{\beta ^2} + 1 - {\eta \over 4}$ and so combining with $(\clubsuit )$ , we have

\begin{align*}|C_j^{t + 1}| \geqslant |C_j^t| \cdot \left(1 - {\eta \over 4} - 2{\beta ^2} - 2\sqrt \varepsilon \right)\gt {\beta ^8}|C_j^t|,\end{align*}

as needed. Then we can assume that $\chi (\lfloor {{i \over 2}} \rfloor ,\lfloor {{j \over 2}} \rfloor ) = \chi (\lfloor {{{i + 1} \over 2}} \rfloor ,\lfloor {{j \over 2}} \rfloor ) = {\rm{blue}}$ . Then by definition, we can see that $(\lfloor {{j \over 2}} \rfloor ,\lfloor {{i \over 2}} \rfloor ),(\lfloor {{j \over 2}} \rfloor ,\lfloor {{{i + 1} \over 2}} \rfloor ) \in E(D)$ , which implies that $\pi (\lfloor {{j \over 2}} \rfloor )\lt \pi (\lfloor {{i \over 2}} \rfloor),$ $ \pi (\lfloor {{{i + 1} \over 2}} \rfloor )$ . Therefore, $(\lfloor {{j \over 2}} \rfloor ,\lfloor {{j \over 2}} \rfloor + 1)$ appears before $\lfloor {{i \over 2}} \rfloor ,\lfloor {{i \over 2}} \rfloor + 1$ in the sequence $(i_1, i_1+1), \dots , (i_{k-1}, i_{k-1}+1)$ which in turn implies that $j \in I_t,$ contradicting our assumption.

We have shown that we can run all $k-1$ steps of the above procedure. Finally, we have two candidate sets $C^k_1, C^k_{2k}$ of size at least $\beta ^{8k} \gamma _2 |B_1|$ into which we can embed the vertices $x_1$ and $x_{2k}$ respectively. Recalling that $d({C_1},{C_{2k}})\lt 1 - {\eta \over 4}$ , by $\varepsilon _1$ -regularity, there are vertices $u \in C^k_1, v \in C^k_{2k}$ such that $uv \not \in E(G).$ Setting $x_1 = u, x_{2k} = v,$ we have that $x_1, x_2, \dots , x_{2k}$ is a monotone induced path in $G$ , contradicting our assumption.

This finishes the proof of Theorem 3.2.

6.1. $B_2$ property implies $B_k$ property in hereditary graphs

Proof of Theorem 1.9. Assume that $\mathcal{G}$ is a hereditary family which has the $B_2$ property with function $f$ . For any $\varepsilon \gt 0$ , we set a function

\begin{align*}h(\varepsilon )=\delta \ll \varepsilon _0\ll \varepsilon _1\ll \varepsilon _2\ll \varepsilon _3\ll \varepsilon _4\ll \varepsilon .\end{align*}

We will show that $\mathcal{G}$ has the $B_k$ property with respect to the function $h$ .

Let $G$ be an $n$ -vertex graph in the family $\mathcal{G}$ such that $G$ has at least $\varepsilon n^k$ $k$ -cliques. Let $m = {4 \over \varepsilon }$ . If $n\leqslant m$ , then we are done by Nikiforov’s result [Reference Nikiforov27]. Otherwise, by Lemma 2.1, there exists $M=M(m,\varepsilon _0)$ such that $V(G)$ admits an $\varepsilon$ -regular equitable partition $\mathcal{P}$ with $K+1$ parts $V_{0},V_{1},\ldots ,V_{K}$ , where $m \leqslant K \leqslant M$ . Setting $|V_1|=q$ , we can see that $(1 - {\varepsilon _0}){n \over K}\leqslant q\leqslant {n \over m}$ .

We then construct an auxiliary graph $G'$ with vertex set $\{V_{1},V_{2},\ldots ,V_{K}\}$ , and a pair of vertices $V_{i},V_{j}$ is adjacent in $G'$ if and only if $(V_i,V_j)$ is $\varepsilon _0$ -regular and $d({V_i},{V_j})\gt {\varepsilon \over 2}$ .