Proof Let $\mathcal {M}=\{(\chi , F): \chi \in \mathcal {C}, F\in \mathcal {F}\}$ . For each $i\in [n]$ , let $C_i$ denote the number of full chains $\chi $ that contain exactly i members of $\mathcal {F}$ . Then, $|\mathcal {M}|=\sum _{i = 1}^n iC_i$ . On the other hand, for each $F\in \mathcal {F}$ , the number of full chains in $\mathcal {B}_n$ that contain F is exactly $\frac {n!}{\binom {n}{|F|}}$ . Hence,

$$\begin{align*}\sum_{i = 1}^n iC_i=\sum_{F\in \mathcal{F}} \frac{n!}{{n\choose |F|}}=\mu(\mathcal{F})n!\ge (q-1+\varepsilon)n!.\end{align*}$$

Clearly, $\sum _{i<q} iC_i\leq (q-1)n!$ . Hence, $|\mathcal {T}|=\sum _{i\geq q} iC_i\geq \varepsilon n!$ .

For the second statement, suppose $|\mathcal {F}|\geq (q-1+\varepsilon ) \max _{F\in \mathcal {F}}\binom {n}{|F|}$ . Then,

$$\begin{align*}\mu(F)=\sum_{F\in \mathcal{F}}\frac{1}{\binom{n}{|F|}}\geq \frac{|\mathcal{F}|}{\max_{F\in \mathcal{F}} \binom{n}{|F|}} \geq q-1+\varepsilon.\end{align*}$$

So, the statement follows from the first statement.

Proof Consider any $\chi \in \mathcal {C,}$ where $\mathcal {T}(\chi ) \neq \emptyset $ . Because $\mathcal {T}$ is q-strong, we have $|\mathcal {T}(\chi )|\geq q$ by definition. Note that for each $F\in \mathcal {T}(\chi )$ that is not among the largest $i-1$ members or smallest $q-i-1$ members in $\mathcal {T}(\chi )$ , there exists a q-chain in $\binom {\mathcal {T}(\chi )}{q}$ that contains F as the ith member. Hence, $|\mathcal {L}^i(\mathcal {M},\chi )|\geq \frac {1}{q}|\mathcal {T}(\chi )|$ and therefore

(4.3) $$ \begin{align} \sum_{\chi\in \mathcal{C}} |\mathcal{L}^i(\mathcal{M},\chi)|\geq \frac{1}{q} \sum_{\chi\in \mathcal{C}} |\mathcal{T}(\chi)|\geq \frac{\varepsilon}{q} n!. \end{align} $$

On the other hand, for each $F\in \mathcal {L}^i(\mathcal {M})$ , F is contained in exactly $n!/\binom {n}{|F|}$ full chains of $\mathcal {B}_n$ and hence belongs to $\mathcal {L}^i(\mathcal {M},\chi )$ for at most $n!/\binom {n}{|F|}$ different $\chi $ . Hence,

(4.4) $$ \begin{align} \sum_{\chi\in \mathcal{C}} |\mathcal{L}^i(\mathcal{M},\chi)|\leq \sum_{F\in\mathcal{L}^i(\mathcal{M})}\frac{n!}{\binom{n}{|F|}}\leq n!\cdot \frac{|\mathcal{L}^i(\mathcal{M})|}{\min_{F\in \mathcal{F}} \binom{n}{|F|}}. \end{align} $$

Combining (4.3) and (4.4), we get $|\mathcal {L}^i(\mathcal {M})|\geq \frac {\varepsilon }{q} \min _{F\in \mathcal {F}} \binom {n}{|F|}$ .

Proof of Lemma 4.9

We begin by explicitly defining for each $\chi \in \mathcal {C}_1^j(\downarrow )$ , and will refer to this vector as the lower-bad profile for $\chi $ relative to $\mathcal {M}^{j - 1}$ .

Fix some $\chi \in \mathcal {C}_1^j(\downarrow )$ . Let $F_1 \supset F_2 \dots \supset F_t$ be the members of $\mathcal {T}^0(\chi )$ . Let $F_{a_1} \supset F_{a_2} \supset \dots \supset F_{a_r}$ be the subsequence of $F_1, \dots F_t$ consisting of all of the members in $\mathcal {T}^{j - 1}(\chi )$ . By definition of ${\mathcal C}_1^j(\downarrow )$ , we have $|B^{j -1}(\chi , \downarrow ) |> |\mathcal {T}^{j-1}(\chi )|/\Delta $ . By the pigeonhole principle, there exist some $i \leq q$ such that

$$\begin{align*}|B^{j-1}(\chi, i, \downarrow)|> |\mathcal{T}^{j-1}(\chi)| /q\Delta.\end{align*}$$

Fix such an i. We will now greedily build a tuple of integers such that for all $1 \leq \ell \leq m$ , $F_{b_\ell }$ is $(i, \delta )$ -lower bad relative to $\mathcal {M}^{j-1}$ and $F_{b_\ell '} \in \mathcal {W}(F_{b_{\ell }}, i, \mathcal {T}^{ j -1}[q])$ , as follows.

Let d be the smallest integer such that $F_{a_d}\in B^{j-1}(\chi , i, \downarrow )$ ; such a d exists because $B^{j-1}(\chi , i, \downarrow ) \neq \emptyset $ . By definition, $F_{a_d}$ is the ith member of some marked chain $(\chi , Q)$ in $\mathcal {M}^{j - 1}(\chi )$ .

In particular, this implies that there are at least $q-i$ additional members of $\mathcal {T}^{j-1}(\chi )$ on $\chi $ below $F_{a_d}$ . Because $F_{a_d}\in B^{j-1}(\chi ,i,\downarrow )$ , it is the ith member of some q-marked chain in $\mathcal {M}^{j - 1}(\chi ) = \binom {\mathcal {T}^{ j - 1}(\chi )}{q}$ , and in particular, there exists at least $i - 1$ members which come before it, and at least $q - i$ members coming after in inside $\mathcal {T}^{ j - 1}(\chi )$ . By property $(b)$ of witness sets, we have then $\{F_{a_{d +1}}, \dots F_{a_{d+ q - i}}\}\cap \mathcal {W}(F_{a_d}, i, \mathcal {T}^{( j - 1)}[q])\neq \emptyset $ . Let $b_1 = a_d$ and let $b_1'$ be the index of the any member $\{F_{a_{d +1}}, \dots F_{a_{d+ q - i}}\}\cap \mathcal {W}(F_{a_d},i, \mathcal {T}^{( j - 1)}[q])$ .

Now, let $d'$ be the smallest integer such that $F_{a_{d'}} \in B^{j-1}(\chi , i, \downarrow )$ and $a_{d'}> b_1'$ if it exists. Just as before, we are guaranteed one of $F_{a_{d'+ 1}}, \dots F_{a_{d'+ q - i}}$ belongs to $\mathcal {W}(F_{a_{d'}}, i, \mathcal {T}^{( j - 1)}[q])$ , and we let $b_2 = a_{d' }$ and $b_2'$ be the index of the any member of $\mathcal {W}(F_{a_{d'}}, i, \mathcal {T}^{( j - 1)}[q]) \cap \{ F_{a_{d'+ 1}}, \dots F_{a_{d'+ q - i}}\} $ . We continue to repeat the process, e.g., by defining $d"$ to be the smallest integer such that $F_{a_{d"}} \in B^{j-1}(\chi , i, \downarrow )$ and $a_{d"}> b_2'$ , if it exists, until no more choices remain. Note that the process goes on at least $\frac {1}{q - i} | B^{j - 1}(\chi , i, \downarrow )|$ many steps, as between $F_{b_i}$ and $F_{b_i'}$ there are at most $q - i$ members of $B^{j - 1}(\chi , i, \downarrow )$ . Furthermore, .

Proof By our assumption of $\chi \in {\mathcal C}_1^j(\downarrow )$ , we must have $\chi \in {\mathcal C}^{\ell }_2$ for $\ell =0,1,\dots , j-1$ and $\chi \in {\mathcal C}_1^j(\downarrow )$ . Therefore, for $1 \leq \ell \leq j - 1$ , we know that $|\mathcal {T}^{\ell }(\chi ) |\geq (1 - \frac {2}{\Delta })|\mathcal {T}^{\ell -1}(\chi )|$ , and hence $|\mathcal {T}^0(\chi )| \leq (\frac {\Delta }{\Delta - 2})^{j}|\mathcal {T}^{j - 1}(\chi )|$ . By definition of $\chi \in \mathcal {C}_{1}^j(\downarrow )$ , we know that $|B^{j - 1}(\chi , \downarrow ) | \geq |{\mathcal T}^{j-1}(\chi )|/\Delta $ . By our choice of i, $|B^{j - 1}(\chi , i, \downarrow ) |\geq \frac {1}{q} |B^{j - 1}(\chi , \downarrow )|$ . By our earlier observation, we have that $m \geq \frac {1}{q - i} | B^{j - 1}(\chi , i, \downarrow )|$ . It, therefore, follows that

$$ \begin{align*}|\mathcal{T}^0(\chi)|\leq \left(\frac{\Delta}{\Delta-2}\right)^{|P|} \Delta |B^{j - 1}(\chi, \downarrow)| \leq \Delta q^2 \left( \frac{\Delta}{\Delta - 2} \right)^{|P|} m\leq K m.\\[-34pt]\end{align*} $$

To prove the last part of the lemma, we will use the following technical result, where here we roughly think of $\mathcal {S}$ as the F which are $(i,\delta )$ -lower bad and $\mathcal {W}^*(F)=\mathcal {W}(F,i,\mathcal {T}^{j-1}[q])$ .

Proof Let $A_\ell $ be the event that $\chi _{\mathcal {F}}(c_\ell ) \in \mathcal {S}$ and $\chi _{\mathcal {F}}(c_\ell ') \in \mathcal {W}^*(\chi _{\mathcal {F}}(b_\ell ))$ . Then,

$$ \begin{align*} \operatorname{\mathrm{Prob}}[A_1\cap A_2 \cap \dots \cap A_m] &= \operatorname{\mathrm{Prob}}[A_1] \operatorname{\mathrm{Prob}}[A_2 | A_1] \dots \operatorname{\mathrm{Prob}}[A_m | A_1\cap A_2 \cap \dots\cap A_{m - 1}]. \end{align*} $$

Note that

$$ \begin{align*}\operatorname{\mathrm{Prob}} [\chi_{\mathcal{F}}(b_\ell') \in \mathcal{W}^*(F) | (\chi_{\mathcal{F}} (c_\ell) &= F)\cap A_1\cap A_2\cap \dots \cap A_{ \ell - 1} ]\\ &= \operatorname{\mathrm{Prob}} [\chi_{\mathcal{F}}(b_\ell') \in \mathcal{W}^*(F) | \chi_{\mathcal{F}} (c_\ell) = F ] ,\end{align*} $$

since conditioned on $\chi _{\mathcal {F}} (c_\ell ) = F $ the events $\chi _{\mathcal {F}}(c_i') \in \mathcal {W}^*(F)$ and $A_1\cap A_2 \cap \dots A_{\ell -1}$ are independent.

Thus, by definition of $\mathcal {W}^*(F)$ , we have

$$ \begin{align*} \operatorname{\mathrm{Prob}} [\chi_{\mathcal{F}}(b_\ell') \in \mathcal{W}^*(F) | \chi_{\mathcal{F}} (c_i) = F) ] &\leq \operatorname{\mathrm{Prob}}[ \chi \cap \mathcal{W}^*(F) \neq \emptyset | \chi_{\mathcal{F}} (c_\ell) = F)]\\ &\leq \delta. \end{align*} $$

Observe that $A_\ell $ is exactly the union over the events $\chi _{\mathcal {F}} (c_\ell ) = F$ over $F \in \mathcal {S}$ and $\chi _{\mathcal {F}}(c_\ell ') \in \mathcal {W}^*(F)$ . Since these events are disjoint, we have that the following series of inequalities hold:

$$ \begin{align*} \operatorname{\mathrm{Prob}}[A_\ell| A_1 \cap A_2 \cap \dots \cap A_{ \ell - 1}] &= \sum_{F \in \mathcal{S}} \Big( \operatorname{\mathrm{Prob}}[\chi_{\mathcal{F}} (c_\ell) = F | A_1\cap A_2 \cap \dots A_{ \ell - 1}] \\ & \cdot \operatorname{\mathrm{Prob}} [\chi_{\mathcal{F}}(c_\ell') \in \mathcal{W}^*(F) | (\chi_{\mathcal{F}} (c_i) = F)\cap A_1\cap A_2\cap \dots A_{ \ell - 1} ] \Big)\\ &\leq \sum_{F \in \mathcal{S}} \operatorname{\mathrm{Prob}}[\chi_{\mathcal{F}} (c_\ell) = F | A_1 \cap A_2 \cap \dots A_{ \ell - 1}] \delta \\ &\leq \delta. \end{align*} $$

Therefore,

$$ \begin{align*} \operatorname{\mathrm{Prob}}[A_1\cap A_2 \dots A_m] &= \operatorname{\mathrm{Prob}}[A_1] \operatorname{\mathrm{Prob}}[A_1 | A_2] \dots \operatorname{\mathrm{Prob}}[A_m | A_1\cap A_2 \cap \dots A_{m - 1}] \\ &\leq \delta^m.\\[-34pt] \end{align*} $$

We now prove the last part of the lemma. Note that for each chain $\chi \in \mathcal {C}_1^j(\downarrow ),$ which satisfies , all of the sets of the form $\chi _{\mathcal {F}}(c_\ell )$ are $(i, \delta )$ -lower bad for some $i\in [q]$ with $\chi _{\mathcal {F}}(c^{\prime }_\ell )$ in the corresponding set $\mathcal {W}(F,i,\mathcal {T}^{j-1}[q])$ . Thus, applying Lemma 4.12 with $\mathcal {S}=\bigcup _{\chi \in \mathcal {C}_1^j(\chi )} B^{j-1}(\chi ,i,\downarrow )$ the set of $(i, \delta )$ -lower bad members, and taking $\mathcal {W}^*(F) = \mathcal {W}(F, i, \mathcal {T}^{j - 1}[q])$ , we have that $\chi $ satisfies the conclusion of Lemma 4.12. Thus, there are no more than $\delta ^m n!$ such chains. Summing over all i, there are at most $q \delta ^m n!$ many chains satisfying .

Proof Let be the function from Lemma 4.9.

Recall that Lemma 4.9.3 states that for all $m\in [n]$ , each bad profile of length $2m$ is a member of $\binom {[Km]}{2m}$ . This together with part Lemma 4.9.4 implies

provided that $\delta < \frac {1}{2^{K + 2} K}$ . This will be less than $\frac {\varepsilon }{18 |P|}$ provided that $\delta < \frac {\varepsilon }{18 |P|2^{ K + 1} K q}$ .

Proof of Theorem 4.7

For each $j\in |P|$ , when building $\mathcal {T}^{j}$ from $\mathcal {T}^{j-1}$ we made sure to remove all the members that are either $(i, \delta )$ -lower-bad or $(i, \delta )$ -upper-bad relative to $\mathcal {T}^{j-1}[q]$ for any $i\in [q]$ . So every member F on a q-chain in $\mathcal {T}^j[q]$ is $\delta $ -robust with respect to $\mathcal {T}^{j-1}[q]$ .

We will inductively prove for all $0\le j\le |P|$ that

$$\begin{align*}\mathcal{T}^j \text{ is } q\text{-strong and }|\mathcal{T}^j|\ge \left(1-\frac{1}{3|P|}\right)^{j}|\mathcal{T}^0|.\end{align*}$$

The base case holds by our conditions. Assume that we have proven the statements for all $\ell \leq j-1$ , which in particular means $|\mathcal {T}^{j-1}|\ge \frac {2}{3}\varepsilon n!$ since $|\mathcal {T}^0|\ge \varepsilon n!$ by hypothesis. Consider any $j\in [|P|]$ . Let $\chi $ be a full chain in $\mathcal {B}_n$ with $\mathcal {T}^j(\chi )\neq \emptyset $ . Then, by our process, this means that $\chi \in \mathcal {C}^\ell _2$ for each $\ell =1,\dots , j-1$ . If $\mathcal {T}^j(\chi )=\mathcal {T}^{j-1}(\chi )$ , then by induction hypothesis, $|\mathcal {T}^j(\chi )|\geq q$ . Hence, we may assume that $\mathcal {T}^{j-1}(\chi )$ contains at least one bad member. If $|\mathcal {T}^{j-1}(\chi )|<\Delta $ , then $\chi $ would have been in $\mathcal {C}^j_1$ , a contradiction. So $|\mathcal {T}^{j-1}(\chi )|\geq \Delta $ . By definition of $\chi \in \mathcal {C}^j_2$ , we have by Lemma 4.8 that

$$\begin{align*}|\mathcal{T}^j(\chi)|\geq \left(1-\frac{2}{\Delta}\right)|\mathcal{T}^{j-1}(\chi)|\geq \Delta-2\geq q.\end{align*}$$

Thus, $\mathcal {T}^{(j)}$ is q-strong. Furthermore,

$$ \begin{align*} |\mathcal{T}^j|&= \sum_{\chi \in \mathcal{B}_n}|\mathcal{T}^j(\chi)| \geq \sum_{\chi \in \mathcal{B}_n}|\mathcal{T}^{j - 1}(\chi)| - \sum_{\chi \in \mathcal{C}_2^j} |B^{j - 1}(\chi) |- \sum_{\chi \in C_1^j(\downarrow)}|\mathcal{T}^{j - 1}(\chi)| - \sum_{\chi \in C_1^j(\uparrow)}|\mathcal{T}^{j - 1}(\chi) |\\ &\geq \sum_{\chi \in \mathcal{B}_n}|\mathcal{T}^{j - 1}(\chi)| - \frac{2}{\Delta} \sum_{\chi \in \mathcal{B}_n}|\mathcal{T}^{j - 1}(\chi)| - \frac{\varepsilon}{9 |P|} n! \quad \text{ (By Lemmas~}4.13 \text{ and }4.14)\\ &\geq|\mathcal{T}^{ j - 1}| - \frac{1}{3 |P|}|\mathcal{T}^{ j - 1}|, \end{align*} $$

where in this last step used $\Delta \ge 12|P|$ and that inductively $\frac {2}{3} \varepsilon n!\leq |\mathcal {T}^{j-1}|$ .

Thus, we have for all $1 \leq j \leq |P|$ ,

$$ \begin{align*}|\mathcal{T}^{j}| \geq \left(1 - \frac{1}{3|P|}\right)^{|P|} |\mathcal{T}^{0}| \geq \frac{2}{3} |\mathcal{T}^{0}|\ge \frac{2 \varepsilon}{3}n!,\end{align*} $$

completing the proof.

Proof By Lemma 4.2, there exists a q-strong 1-marked family $\mathcal {T}^0$ from $\mathcal {F}$ with $|\mathcal {T}^0|\ge \varepsilon n!$ . We can thus apply Theorem 4.7 with this $\mathcal {T}^0$ to obtain a nested sequence $\mathcal {T}^{|P|}\subseteq \cdots \subseteq \mathcal {T}^0$ of q-strong 1-marked chain families such that property 1 holds for $\mathcal {M}^j:=\mathcal {T}^j[q]$ . Moreover, Theorem 4.7 guarantees $|\mathcal {T}^j|\ge |\mathcal {T}^{|P|}|\ge \frac {2\varepsilon }{3} n!$ , so applying Lemma 4.3 to $\mathcal {T}^j$ gives property 2, completing the proof.

Proof We will only prove the case $i<s$ , the other case is analogous. Because F is $(i,s,\gamma )$ -bad, there exist two sets $\mathcal {W}_1,\mathcal {W}_2$ satisfying the conditions in the definition and we will choose $\mathcal {W}_2$ to be minimal, i.e., no $\mathcal {W}^{\prime }_2 \subsetneq \mathcal {W}_2$ satisfies condition three with $\mathcal {W}_1$ . Observe that this minimality of $\mathcal {W}_2$ implies that every $D\in \mathcal {W}_2$ is in the sth position of some q chain in $\mathcal {M}(F, i)$ .

We will show that F is $(i, \delta )$ -lower bound with respect to $\mathcal {M}$ with $W:=\mathcal {W}_2 \cup D^*(F, \mathcal {W}_1)$ being an $(i,\delta )$ -lower witness. Since $\mathcal {M}$ is $\ell $ -gapped and for all $D\in \mathcal {W}_2$ , D is in the sth position of some q-chain in $\mathcal {M}(F, i)$ , we have $|F-D|\geq \ell (s-i)$ . Since n is sufficiently large and $F,D\in \widetilde {\mathcal {B}}_n$ , $|F-D|\leq n/6$ . Thus, for a uniformly chosen random full chain $\chi \in \mathcal {C,}$ we have

$$ \begin{align*} \operatorname{\mathrm{Prob}}[\chi \cap \mathcal{W}_2 \neq \emptyset | F \in \chi] &\leq \sum_{D \in \mathcal{W}_2} \frac{1}{\binom{|F|}{|F - D|}} & \text{ by~}(4.5)\\ &\leq |\mathcal{W}_2| \frac{1}{\binom{n / 3}{ \ell(s - i)}}\\ &\leq \gamma n^{\ell (s - i)} (3\ell(s - i))^{\ell (s - i)} n^{ - \ell(s - i)}\\ &\leq \gamma (3\ell(s - i))^{\ell (s - i)}\\ &\leq \frac{\delta}{2}, \end{align*} $$

by the choice of $\gamma = \frac {\delta }{2 (3 \ell q)^{\ell q}}$ . On the other hand by Proposition 4.16,

$$ \begin{align*}\operatorname{\mathrm{Prob}}[\chi \cap D^*(F, \mathcal{W}_1) \neq \emptyset | F \in \chi] \leq \frac{39 |P| \sqrt{n \ln n }}{n} \leq \frac{\delta}{2}.\end{align*} $$

Thus, by the union bound,

$$ \begin{align*}\operatorname{\mathrm{Prob}}[ \chi \cap \mathcal{W} \neq \emptyset | F \in \chi] \leq \delta.\end{align*} $$

This implies that F is $(i, \delta )$ -lower bad, because by definition, every $(\chi , Q)\in \mathcal {M}(F, i)$ intersects W.

Proof Because P is a tree poset, there exists an ordering $v_1, v_2, \dots v_{|P|}$ of the members of P with $v_1 = v$ such that every member $v_j$ with $j\ge 2$ has exactly one neighbor $v_{j'}$ with $j'<j$ in the Hasse diagram of P, and we call this $v_{j'}$ the parent of $v_j$ .

Apply Theorem 4.15 to $\mathcal {F}$ with $q, \varepsilon $ to find a $\delta = \delta (\varepsilon , q)> 0$ and a nested sequence of q-marked chains $\mathcal {M}^{|P|}\subseteq \mathcal {M}^{|P|-1}\subseteq \cdots \subseteq \mathcal {M}^0$ with markers from $\mathcal {F}$ such that for all $j\in [|P|],$ we have that for each $(\chi ,Q)\in \mathcal {M}^j$ and $F\in Q$ , we have that F is $\delta $ -robust with respect to $\mathcal {M}^{j-1}$ .

Fix $\gamma $ be obtained from Lemma 4.20 applied with $q, \ell , \delta $ .

Fix $F_0\in \mathcal {L}^{\mathrm {r}(v_1)}(\mathcal {M}^{|P|})$ and define our initial embedding $\varphi ^1(v_1):=F_0$ . To define the final embedding, we iteratively extend $\varphi ^j: \{ v_1, v_2, \dots , v_j \} \rightarrow \mathcal {F} $ to $\varphi ^{j+1}$ maintaining the following properties:

1. (C₁): Let $H_j(P) := H(P)[\{v_1, v_2, \ldots , v_j\}]$ , that is, the Hasse diagram induced by the first j elements of P. For all edges $v_av_b$ in $H_j(P)$ , there exists a $(\chi , Q)$ in $\mathcal {M}^{|P| - j}$ with $\varphi ^{j}(v_a)$ in the $\mathrm {r}(v_a)$ th position of Q and $\varphi ^{j}(v_b)$ in the $\mathrm {r}(v_b)$ th position of Q.

2. (C₂): For all noncomparable pairs $v_a, v_b \in \{v_1, \dots , v_j\}^2$ , $\varphi ^j(v_a)$ and $\varphi ^j(v_b)$ are not comparable in $\mathcal {B}_n$ .

3. (C₃): $\{ \varphi ^j(v_1), \varphi ^j(v_2), \dots \varphi ^j(v_j)\} \not \in \Gamma $ .

Note that these properties are satisfied for $\varphi ^1$ , since $(C_1)$ and $ (C_2)$ both hold vacuously and $(C_3)$ holds because $F_0 \not \in \Gamma $ by Definition 5.1 part 1.

Let us emphasize that the choice of $F_0$ and $(C_1)$ ensures $\varphi ^j(v_c) \in \mathcal {L}^{\mathrm {r}(v_c)}(\mathcal {M}^{|P| - j})$ for all $v_c \in \{v_1, \dots v_j\}$ . This will be useful later.

Furthermore, $(C_1)$ and $(C_2)$ ensure the poset induced by $\varphi ^{j + 1}(v_1), \varphi ^{j }(v_2), \dots , \varphi ^{j }(v_{j})$ has Hasse diagram isomorphic to $H_{j}(P)$ . As two posets are isomorphic if and only if their Hasse diagrams are isomorphic, by the end of the process, we will obtain an induced copy $P'$ of P. Let us now show that indeed these partial embeddings are possible to construct, and count how many choices we have at each step.

Now, let $j\geq 1$ and suppose $(C_1)$ – $(C_3)$ hold for $\varphi ^j$ . We want to extend $\varphi ^j$ to a partial embedding $\varphi ^{ j + 1}$ so that $(C_1)$ – $(C_3)$ hold for $\varphi ^{ j + 1}$ .

For notational convenience let $x:=v_{j + 1}$ , and let y denote the unique parent of x which has already been embedded. We wish to find at least one way of embedding x. For further convenience, we only consider the case $y>_P x$ , the other case being analogous. This, in particular, implies $\mathrm {r}(y)<\mathrm {r}(x)$ .

Recall the formal definition of $D^*(F, \mathcal {S})$ from Section 4.3 for a member $F\in \mathcal {F}$ and a family $\mathcal {S}$ which is simply the set of all members $S\in \mathcal {S,}$ which are downsets of F and are comparable to some set of $\mathcal {S}$ . Furthermore, let $\widehat {\mathcal {M}}^{|P| - j - 1} := \mathcal {M}^{|P| - j - 1}(\varphi ^j(y), \mathrm {r}(y))$ , that is the family of all q-marked chains in $\mathcal {M}^{|P| - j - 1}$ , which have $\varphi ^j(y)$ in their $\mathrm {r}(y)$ th position. Let $\mathcal {P}_j = \{ \varphi ^j(z) : z \not \geq _P y , z \in \{ v_1, v_2, \dots , v_{j} \}\}$ , that is, the set of all images of currently embedded elements of P which are either below y or incomparable with y, and thus not cannot be comparable with $\varphi ^{j + 1}(x)$ because in the embedding $\varphi ^j$ , x has the unique neighbor y.

The following set encodes the choices for x that would preserve $\varphi ^{j+1}$ being an induced homomorphism. Now, let

$$ \begin{align*}\mathcal{A} =\mathcal{L}^{\mathrm{r}(x)}(\widehat{\mathcal{M}}^{|P| - j - 1}) - D^*(\varphi^j(y), \mathcal{P}_j ),\end{align*} $$

i.e., this is the set of F which are not in the forbidden neighborhood of $\mathcal {P}_j$ and which are in the $\mathrm {r}(x)$ th position of some q-marked chain of $\mathcal {M}^{|P| - j - 1}$ , which has $\varphi ^j(y)$ in the $\mathrm {r}(y)$ th position.

Claim

$$ \begin{align*}|\mathcal{A}| \geq \gamma n^{\ell (\mathrm{r}(x) - \mathrm{r}(y)).}\\[-34pt]\end{align*} $$

Let $\mathcal {A}' = \mathcal {A} - \{F \in \mathcal {F}: \{F, \varphi ^j(v_1), \varphi ^{j}(v_2), \dots \varphi ^{j}(v_j)\} \in \Gamma \}$ . By $(C_3)$ and Definition 5.1, $|\mathcal {A}'| \geq \frac {1}{2} \gamma n^{\ell (\mathrm {r}(x) - \mathrm {r}(y))}$ . Fix one such $F \in \mathcal {A}'$ . We claim that by embedding x into any set in $\mathcal {A}'$ , we will extend $\varphi ^j$ to $\varphi ^{j+1}$ , satisfying all the desired properties $(C_1)$ – $(C_3)$ . Indeed, define $\varphi ^{j + 1}(v_{a}) = \varphi ^{j} (v_{a})$ for all $a \in [j]$ and $\varphi ^{ j + 1}(x) = F$ . Let us now check that $\varphi ^{j+1}$ satisfies $(C_1)$ – $(C_3)$ .

Since $\mathcal {M}^{ |P| - j} \subseteq \mathcal {M}^{|P| - j - 1}$ , for all $a, b \in [j]$ , $(C_1)$ and $(C_2)$ hold for all such pairs so we only need to check that these two conditions are satisfied for pairs of form $(x, v_a)$ for any $a\in [j]$ .

By definition of $\mathcal {A}$ , $\varphi ^{j + 1}(x)$ is in the $\mathrm {r}(x)$ th position of some $(\chi , Q) \in \mathcal {M}^{|P| - j -1}$ , which has $\varphi ^{ j + 1}(y)$ in the $\mathrm {r}(y)$ th position.

Let $a \in [j]$ , and suppose $v_a$ is not comparable with x. We would like to show that $\varphi ^{j+1}(v_a)$ and F are incomparable. It is easy to see that $v_a \not \geq y$ , so $\varphi ^{j + 1}(v_a) = \varphi ^j(v_a) \in \mathcal {P}_j$ . Since $F \in D(\varphi ^j(y))\setminus D^*(\varphi ^j(y), \mathcal {P}_j)$ by our choice, F is not comparable with any element in $\mathcal {P}_j$ , in particular, $\varphi ^{j + 1}(v_a)$ .

To see $\varphi ^{ j + 1}$ satisfies $(C_3)$ , notice that F was chosen so that $\{F, \varphi ^j(v_1), \varphi ^{j}(v_2), \dots \varphi ^{j}(v_j)\} \not \in \Gamma $ .

Finally, to check that the desired counting bound does hold, it is enough to observe that at every step while going from $\varphi ^j$ to $\varphi ^{j+1}$ starting at $j\geq 2$ we had at least $\frac {1}{2} \gamma n^{\ell (\mathrm {r}(x) - \mathrm {r}(y))}$ many choices to embed $x=v_{j+1}$ , where y is the parent of x among already embedded vertices $v_1, v_2, \dots , v_{j}$ . Taking the product over all $j=2, \dots , |P|-1$ gives us the desired result.

Proof Let v be any vertex of P and $\gamma = \gamma (\varepsilon , q, \ell )$ the real number returned by Theorem 5.2 applied with $\varepsilon , q, \ell $ .

Assume on the contrary that $\mathcal {F}$ contains less than

$$\begin{align*}\left( \frac{\gamma}{2} \right)^{|P| - 1}\left(\prod_{xy\in H(P)}n^{\ell |\mathrm{r}(y)-\mathrm{r}(x)|} \right)\cdot N,\end{align*}$$

induced copies of P.

Let $\mathcal {F}_{\mathrm {b}}$ be the set of elements in $\mathcal {F}$ which are embedded as v in more than $\left (\frac {\gamma }{2} \right )^{|P| - 1} \prod _{xy\in H(P)}n^{\ell |\mathrm {r}(y)-\mathrm {r}(x)|}$ induced copies of P.

Since by assumption, we know that

$$ \begin{align*}|\mathcal{F}_{\mathrm{b}} | \cdot\left( \frac{ \gamma}{2} \right)^{|P| - 1} \prod_{xy\in H(P)}n^{\ell |\mathrm{r}(y)-\mathrm{r}(x)|} \leq \left(\frac{\gamma}{2} \right)^{|P| - 1} \left(\prod_{xy\in H(P)}n^{\ell |\mathrm{r}(y)-\mathrm{r}(x)|} \right)\cdot N,\end{align*} $$

$$ \begin{align*}\implies |\mathcal{F}_{\mathrm{b}}| \leq N.\end{align*} $$

Let $\mathcal {F}' = \mathcal {F} - \mathcal {F}_{\mathrm {b}}$ . Then, $\mu (\mathcal {F}') \geq q - 1 + \varepsilon $ by assumption.

Now, apply Theorem 5.2 to $\mathcal {F}'$ with $\Gamma = \emptyset , q, \ell , \varepsilon $ . Then, there is $F \in \mathcal {F}$ playing the role of v in at least

$$\begin{align*}\left(\frac{\gamma}{2}\right)^{|P| - 1} \prod_{xy\in H(P)}n^{\ell |\mathrm{r}(y)-\mathrm{r}(x)|},\end{align*}$$

induced copies of P. This contradicts the construction of $\mathcal {F}'$ , and so $\mathcal {F}$ contains at least

$$\begin{align*}\left( \frac{\gamma}{2} \right)^{|P| - 1}\left(\prod_{xy\in H(P)}n^{\ell |\mathrm{r}(y)-\mathrm{r}(x)|} \right) \cdot N,\end{align*}$$

induced copies of P, completing the proof of the theorem.

Proof For ease of notation, let $\mathcal {M}_{n, q}$ be the middle q layers of $\mathcal {B}_n$ . Given a poset homomorphism $\mathrm {r}:P\to [q]$ , we say that a copy $P'$ of P in $\mathcal {M}_{n,q}$ is of type- r if for each $x\in P$ , the member $F_x\in P'$ corresponding to x lies in the $\mathrm {r}(x)$ th level of $\mathcal {M}_{n,q}$ . Observe that every copy of $P \in \mathcal {M}_{n,q}$ must be of type $\mathrm {r}$ for some $\mathrm {r}$ . As such, to prove the result, it will suffice to show that for all homomorphisms $\mathrm {r}$ , the number of induced copies of P or type $\mathrm {r}$ in $\mathcal {M}_{n,q}$ is at most

$$\begin{align*}\prod_{xy\in H(P)}n^{|\mathrm{r}(x)-\mathrm{r}(y)|}\cdot {n\choose n/2}.\end{align*}$$

Let $x_1,\ldots ,x_{|P|}$ be an ordering of P such that for all $j\geq 2 x_j$ has a unique neighbor, called the parent of $x_j$ , among $x_1, \dots , x_{j-1}$ if we view $H(P)$ as an undirected graph. Note that such an ordering exists since P is a tree poset. We can identify the copies of P in $\mathcal {M}_{n,q}$ of type- $\mathrm {r}$ by tuples $(F_1,\ldots ,F_{|P|})$ of members of $\mathcal {M}_{n,q}$ such that the map $f(x_i)=F_i$ defines a copy of P of type $\mathrm {r}$ . So, it is enough to upper bound the number of such tuples.

Since $F_1$ must be a member of the $\mathrm {r}(x_1)$ th layer of $\mathcal {M}_{n,q}$ , the number of choices for $F_1$ is at most ${n\choose n/2}$ . Given that $F_1,\ldots ,F_{j-1}$ have been selected, let $x_i$ be the parent of $x_j$ . If $x_i<x_j$ , then $F_j$ must be a set containing $F_i$ together with $\mathrm {r}(x_i)-\mathrm {r}(x_j)$ additional elements from $[n]$ , and the number of such sets is at most $n^{\mathrm {r}(x_i)-\mathrm {r}(x_j)}$ . Similarly if $x_j<x_i$ then the number of choices for $F_j$ is at most $n^{\mathrm {r}(x_j)-\mathrm {r}(x_i)}$ . Multiplying the number of choices for each step gives the total result, since each edge of $H(P)$ is counted exactly once by some $n^{|\mathrm {r}(x_i)-\mathrm {r}(x_j)|}$ term.

Proof of Theorem 1.3

Using Chernoff bounds, it is standard [Reference Boehnlein and Jiang8, Reference Griggs and Lu21] to show that the number of sets $F\in \mathcal {B}_n$ with $||F|-n/2|>2\sqrt {n\ln {n}}$ is $o({n \choose n/2})$ . Thus, we may assume $\mathcal {F}\subseteq \widetilde {\mathcal {B}}_n$ .

Note that $\mathcal {F}$ is one-gapped and for every subfamily $\mathcal {S}$ of size $N=\frac {\varepsilon }{2} \binom {n}{n / 2}$ , we have that $\mu (\mathcal {F} - \mathcal {S}) \geq q - 1 + \frac {\varepsilon }{2}$ . Thus, applying Theorem 5.4 with $\mathcal {F}, q, \ell = 1, \frac {\varepsilon }{2}$ and taking the maximum over $\mathrm {r} : P \rightarrow [q],$ we have that the number of induced copies of P in $\mathcal {F}$ is at least

$$\begin{align*}\Omega\left(\max_{\mathrm{r}}\prod_{xy\in H(P)}n^{|\mathrm{r}(y)-\mathrm{r}(x)|}\cdot {n\choose n/2}\right)=\Omega\left(\sum_{\mathrm{r}}\prod_{xy\in H(P)}n^{|\mathrm{r}(y)-\mathrm{r}(x)|}\cdot {n\choose n/2}\right)=\Omega\left(M^*(n,q,P)\right),\end{align*}$$

where the first equality holds since there are at most $|P|^q=O_{q,P}(1)$ possible poset homomorphisms $\mathrm {r}:P\rightarrow [q]$ , and the second does because of Lemma 5.5.

Proof We say that $\mathcal {D} \subseteq \mathcal {B}_n$ is saturated if $1 \leq |\mathcal {D}| \leq |P|$ and

$$ \begin{align*}\deg(\mathcal{D}) = \lfloor (\delta n^{\ell} )^{ |P| - |\mathcal{D}|} \rfloor.\end{align*} $$

We say a subfamily $\mathcal {K} \subseteq \mathcal {B}_n$ with $|\mathcal {K}| \leq |P|$ is inadmissible if there exists a subfamily $\mathcal {D} \subseteq \mathcal {K}$ that is saturated. Otherwise, we say $\mathcal {K}$ is admissible.

Observe that proving the lemma is equivalent to saying that there exists an admissible set $\mathcal {K}$ which forms an induced copy of P which is not already in $\mathcal {H}$ . To show this, we will start by removing from $\mathcal {F}$ any F such that $\{F\}$ is saturated, as no such F can ever be used in an admissible $\mathcal {K}$ . For this, we observe the following.

Here and throughout the proof, we will make frequent use of the inequality $\lfloor (\delta n^\ell )^{|P|-i}\rfloor \ge \frac {1}{2} (\delta n^{\ell })^{|P|-i}$ for all $i\le |P|$ , since n is sufficiently large.

Proof of Claim 5.8

We first show that

$$\begin{align*}\lfloor (\delta n^{\ell})^{|P|-1}\rfloor \cdot |\mathcal{F}_{\mathrm{sat}}|\le \sum_{F\in \mathcal{F}_{\mathrm{sat}}} \deg(\{F\})\le |P|\cdot |\mathcal{H}|.\end{align*}$$

Indeed, the lower bound for the sum follows from the definition of what it means for $\{F\}$ to be saturated; the upper bound comes from the fact that the sum counts the number of pairs $(P',F)$ with $P'$ a copy of P in $\mathcal {H}$ and $F\in P'\cap \mathcal {F}_{\mathrm {sat}}$ .

Using the inequality above together with (P1) shows that

$$\begin{align*}\frac{1}{2}(\delta n^{\ell})^{|P|-1}\cdot |\mathcal{F}_{\mathrm{sat}}|\le \lfloor (\delta n^{\ell})^{|P|-1}\rfloor \cdot |\mathcal{F}_{\mathrm{sat}}|\le |P|\cdot |\mathcal{H}|\le |P| \cdot \delta^{|P|} n^{\ell (|P|-1)}{n\choose n/2},\end{align*}$$

and rearranging gives the desired result, by choosing $\delta $ sufficiently small with respect to $\varepsilon , P$ .

Define $\mathcal {F}':=\mathcal {F}-\mathcal {F}_{\mathrm {sat}}$ . Note that

$$\begin{align*}|\mathcal{F}'|\ge (k-1+\varepsilon/2){n\choose n/2}.\end{align*}$$

For every $\mathcal {K}\subseteq B_n$ , let us define

$$ \begin{align*}\mathcal{Z}(\mathcal{K}) : = \{ F \in \mathcal{F}' : \{ F \} \cup \mathcal{K} \text{ is inadmissible} \}.\end{align*} $$

The intuition here is that if we have already partially built some set $\mathcal {K}$ to eventually be used in a copy of P, then $\mathcal {Z}(\mathcal {K})$ represents the set of “bad choices” of F that we could add to $\mathcal {K}$ to make it inadmissible. A simple double counting argument shows that the number of such “bad choices” is relatively small.

Claim For any admissible $\mathcal {K} \subseteq \mathcal {B}_n$ and $|\mathcal {K}| \leq |P|$ , we have

$$ \begin{align*}|\mathcal{Z}(\mathcal{K})| \leq 2^{|\mathcal{K}|} \cdot 2 \delta |P| n^{\ell}.\end{align*} $$

Proof of Lemma 5.9

For every subfamily $\mathcal {D} \subseteq \mathcal {K}$ , let

$$ \begin{align*}\mathcal{S}(\mathcal{D}) := \{ F \in \mathcal{B}_n : \{ F\} \cup \mathcal{D}\text{ is saturated} \}.\end{align*} $$

Observe that since $\mathcal {K}$ is admissible, having $\{F\}\cup \mathcal {K}$ inadmissible implies that the subsets of $\{F\}\cup \mathcal {K}$ that are saturated must be of the form $\{F\}\cup \mathcal {D}$ for some $\mathcal {D}\subseteq \mathcal {K}$ , i.e., we have $\mathcal {Z}(\mathcal {K})=\bigcup _{\mathcal {D}\subseteq \mathcal {K}} \mathcal {S}(\mathcal {D})$ . Moreover, because every $F \in \mathcal {Z}(\mathcal {K}) \subseteq \mathcal {F}'$ is unsaturated, we have that $\mathcal {S}(\emptyset )=\emptyset $ . In total, then we see that

(5.6) $$ \begin{align} \mathcal{Z}(\mathcal{K})=\bigcup_{\mathcal{D}\subseteq \mathcal{K}, \mathcal{D} \neq \emptyset} \mathcal{S}(\mathcal{D}), \end{align} $$

and it now suffices to bound $|\mathcal {S}(\mathcal {D})|$ for each $\mathcal {D} \neq \emptyset $ .

Fix some $\mathcal {D}\subseteq \mathcal {K}$ non-empty. Let $\mathcal {E}$ denote the set of pairs $(P',F),$ where $P'$ is a copy containing $\mathcal {D}$ and $F\in P'$ is arbitrary. Let $\mathcal {E'} \subseteq \mathcal {E}$ be the set of tuples $(P', F) \in \mathcal {E}$ with the additional property that $F \in \mathcal {S} (\mathcal {D})$ . With this, we see

(5.7) $$ \begin{align}\sum_{F\in \mathcal{S}(\mathcal{D})} \deg(\{F\}\cup \mathcal{D})=|\mathcal{E}'|\le |\mathcal{E}|= |P|\deg(\mathcal{D}).\end{align} $$

Since $\{F\}\cup \mathcal {D}$ is saturated for each $F\in \mathcal {S}(\mathcal {D})$ , we have $\deg (\{F\} \cup \mathcal {D})=\lfloor (\delta n^{\ell })^{|P|-|\mathcal {D}|-1} \rfloor $ . Since $\mathcal {D}\subseteq \mathcal {K}$ is unsaturated (because $\mathcal {K}$ is admissible), we have $\deg (\mathcal {D})\leq (\delta n^{\ell })^{|P|-|\mathcal {D}|}$ . Therefore, (5.7) implies

$$ \begin{align*}|\mathcal{S}(\mathcal{D})| \lfloor (\delta n^\ell)^{ |P| - |\mathcal{D}| - 1} \rfloor = \sum_{ F \in \mathcal{S}(\mathcal{D})} \deg(\{F\} \cup \mathcal{D}) \leq |P| \deg(\mathcal{D}) \leq |P| (\delta n^{\ell})^{ |P| - |\mathcal{D}|} .\end{align*} $$

This implies

$$ \begin{align*}|\mathcal{S}(\mathcal{D}) |\leq 2 \delta |P| n^{\ell} .\end{align*} $$

This together with (5.6) gives $|\mathcal {Z}(\mathcal {K})|\leq 2^{|\mathcal {K}|} \cdot 2 \delta |P| n^{\ell }$ as desired.

To complete the proof, let $\gamma = \gamma (\varepsilon , k, \ell )$ be derived from Theorem 5.2 applied with parameters $P, q = k$ , $\ell ,$ and $\frac {\varepsilon }{2}$ . We let $\Gamma $ be the set of all inadmissible sets. Note that by choosing $\delta $ sufficiently small with respect to $\gamma $ , we may ensure that for all $\mathcal {K}$ admissible, $\mathcal {Z}(\mathcal {K}) \leq \frac {\gamma }{2} n^{\ell }$ by Claim 5.9. This guarantees $\Gamma $ to be $(\frac {\gamma }{2}, \ell , \mathcal {F}')$ -bounded family. Now, applying Theorem 5.2 to $\mathcal {F}'$ , $\Gamma ,$ and any $v\in P$ , we obtain an induced copy $P'$ of P such that $\deg (\mathcal {D}) < \lfloor (\delta n^{\ell } )^{|P| - |\mathcal {D}|} \rfloor $ for all $\mathcal {D} \subseteq P'$ , and so $P' \not \in \mathcal {H}$ . Adding $P'$ to our collection $\mathcal {H}$ preserves the desired maximum degree condition.

Proof If $(k - 1 + \varepsilon ) \binom {n}{n / 2} \leq |\mathcal {F}| \leq 3k \binom {n}{n /2}$ , then we apply Theorem 5.6 with $\varepsilon $ , $\ell = 1$ , to find a $\delta _1 := \delta _1(\varepsilon , P)$ and a subgraph $\mathcal {H} \subseteq \mathcal {G}_P[\mathcal {F}]$ with properties:

1. (1) $|\mathcal {H}| \geq \delta _1^{|P|} n^{|P| - 1}\binom {n}{ n / 2}$ ;

2. (2) $\Delta _j(\mathcal {H}) \leq (\delta _1 n)^{|P| - j}$ for all $1 \leq j \leq |P| $ .

Note that this satisfies the conditions of Lemma 6.1, with $c = \frac {2}{\delta _1^{|P|}}$ and $\tau = \tau (\mathcal {F}, k) = \frac {1}{n}$ . Applying Lemma 6.1 to $\mathcal {H}$ then gives the result.

If $|\mathcal {F}| \geq 3k \binom {n}{n / 2}$ , then we arbitrarily split $\mathcal {F}$ into $t + 1$ families $\mathcal {F} = \mathcal {F}_0 \cup \mathcal {F}_1 \cup \dots \cup \mathcal {F}_t$ such that $|\mathcal {F}_0| < 3k \binom {n}{n / 2}$ and each $|\mathcal {F}_i| = 3k \binom {n}{ n / 2}$ . For each $\mathcal {F}_i$ and $j \in \{0, 1, 2\}$ , define

$$ \begin{align*}\mathcal{F}_i^j := \{F \in \mathcal{F}_i : |F| \equiv j \mod 3\}.\end{align*} $$

Since the $\mathcal {F}_i^j$ partition $\mathcal {F}_i$ , there exists some $j,$ where $|\mathcal {F}_i^j| \geq k \binom {n}{n / 2}$ . For this j, let $\mathcal {F}_i' = \mathcal {F}_i^j$ .

Since $\mathcal {F}_i'$ is a $3$ -gapped family, we may apply Theorem 5.6 to $\mathcal {F}_i'$ with $\varepsilon = \frac {1}{2}$ , $\ell = 3$ , to find a $\delta _2 : = \delta _2(\varepsilon , P)$ a subgraph $\mathcal {H}_i \subseteq \mathcal {G}_P[\mathcal {F}_i]$ with properties:

1. (1) $|\mathcal {H}| \geq \delta _2^{|P|} n^{3(|P| - 1)}\binom {n}{ n / 2}$ ;

2. (2) $\Delta _j(\mathcal {H}) \leq (\delta _2 n)^{3(|P| - j)}$ for all $1 \leq j \leq |P| $ .

Consider now the collection $\mathcal {H} = \cup _{i = 1}^t \mathcal {H}_i \subseteq \mathcal {G}_P[\mathcal {F}]$ , noting that because the $\mathcal {H}_i$ are vertex disjoint we have $\Delta _\ell (\mathcal {H})=\max _i \Delta _\ell (\mathcal {H}_i)$ for all $1 \leq b \leq |P|$ , and hence

$$ \begin{align*}\Delta_b(\mathcal{H})= \max_j \Delta_{b}(\mathcal{H}_j)= (\delta_2 n)^{3(|P|-b)} \leq \frac{6k}{\delta_2^{|P| }} n^{3(1-b)}\cdot \frac{t \delta_2^{|P| } n^{3(|P|-1)}{n\choose n/2}}{6k t{n\choose n/2}} \leq \frac{6k}{\delta_2^{|P|}} n^{3(1-b)}\cdot \frac{|\mathcal{H}|}{|\mathcal{F}|},\end{align*} $$

where this last step used that $|\mathcal {F}| \leq 3 k ( t+ 1) {n\choose n/2} \leq 6k t \binom {n}{n / 2}$ by how we defined t and that $|\mathcal {H}|=\sum |\mathcal {H}_j| \geq t\cdot \delta _2^{|P|} n^{3(|P|-1)}{n\choose n/2}$ . With this, we can apply Lemma 6.1 to $\mathcal {H}$ with $c = \frac {6k}{\delta _2^{|P|}}$ and $\tau = \tau (\mathcal {F}, k) = \frac {1}{n^3}$ .

Proof of Theorem 6.4

Let $\delta := \delta (\varepsilon , P)$ be given by Corollary 6.2, and choose K large depending on $\varepsilon , \delta , |P|, k$ . Let n be sufficiently large.

Fix some $I \in \mathcal {I}(\mathcal {G}_P)$ , we will apply Corollary 6.2 a certain number of times, which we will denote by $m = m(I)$ , to construct two sequences $\mathcal {F}_1, \mathcal {F}_2, \dots \mathcal {F}_{m + 1}$ and $T_1, T_2, \dots T_m$ of subsets of $V(\mathcal {G}_P)$ .

First set $\mathcal {F}_1 := \mathcal {B}_n$ . Then apply Corollary 6.2 to $\mathcal {F}_1$ with $\frac {\varepsilon }{2}$ . This gives us a $T(I)$ of size less than $|P| \tau (\mathcal {F}_1, k) |\mathcal {F}_1|$ and a $f(T(I))$ of size less than $(1 - \delta )|\mathcal {F}_1|$ such that $I \subseteq T(I) \cup f(T(I))$ . Let $T_1 = T(I)$ and $\mathcal {F}_2 = f(T(I)) - T(I)$ . In general, as long as $|\mathcal {F}_i| \geq (k - 1 + \frac {\varepsilon }{2})\binom {n}{ n / 2}$ , apply Corollary 6.2 to $\mathcal {F}_i$ with $\frac {\varepsilon }{2}$ . This gives a $T(I)$ and an $f(T(I))$ such that $T(I) \subseteq \mathcal {F}_i \cap I$ and $\mathcal {F}_i \cap I \subseteq f(T(I)) \cup T(I)$ . Let $T_i = T(I)$ , and $\mathcal {F}_{i + 1} = f(T(I)) - T(I)$ .

It is easy to see that the above construction will inductively maintain the following properties for all i:

1. (i) $I \subseteq \mathcal {F}_{i + 1} \cup T_1 \cup T_2 \cup \dots \cup T_i$ ;

2. (ii) $ \mathcal {F}_{i + 1}, T_1, \dots T_i$ are pariwise disjoint;

3. (iii) $\mathcal {F}_{i + 1}$ depends only on $\mathcal {F}_i$ and $T_i$ ;

4. (iv) $|\mathcal {F}_{i + 1}| \leq (1 - \delta ) |\mathcal {F}_i|$ .

We define our fingerprint $(\mathcal {T}, g, C)$ of $\mathcal {G}_P$ by setting

$$ \begin{align*}g(I) := (T_1, T_2, \dots T_m) \text{ and } C(g(I)) := \mathcal{F}_{m + 1} \cup T_1 \cup \dots \cup T_m,\end{align*} $$

and letting $\mathcal {T} := \{ g(I) : I \in \mathcal {I}(\mathcal {G}_P)\}$ . Note that property (iii) implies that C is well defined, as the choice of $\mathcal {F}_{m + 1}$ does not depend on I, while property (i) guarantees that it is a container function. Similarly, (ii) together with how we constructed $T_i$ guarantees that g is a fingerprint function.

In order to check that the constructed fingerprint satisfies the conditions of the theorem, we first bound the sizes of the $T_i$ ’s, and then the number of iterations.

To begin, let $2 \leq m_0 \leq m$ be minimum such that $|\mathcal {F}_{m_0}| \leq 3 k \binom {n}{n / 2}$ and observe that by property (iv) and definition of $\tau (\mathcal {F}, k)$ :

$$ \begin{align*}\tau(\mathcal{F}_i, k) |\mathcal{F}_i| \leq \begin{cases} n^{-3} \cdot 2^n & \text{if } i < m_0, \\ n^{-1} \cdot (1 - \delta)^{ i - m_0} 3k \binom{n}{n / 2} & \text{otherwise}. \end{cases}\end{align*} $$

Since $|\mathcal {F}_i|$ decays at a geometric rate by (iv), we have that $m = O_{\varepsilon , P}(\log n )$ and $m - m_0 = O_{\varepsilon , P}(1)$ .