2.1. Sketch proof for non-spanning trees

Consider the case of Theorem 4(1) in which $Q$ is a tree on $n$ vertices and $G$ is a directed graph on $(1+\alpha )n$ vertices with $\delta ^0(G) \geq (1/2 +\alpha )|G|$ . Let $T$ be an orientation of $Q$ ; our goal is then to embed $T$ within $G$ . In this case we may proceed as follows.

1. (i) Apply the regularity lemma to partition $V(G)$ into clusters $V_1, \ldots , V_k$ of equal size, as well as a small set of exceptional vertices $V_0$ which can safely be ignored, so that almost all pairs of clusters form regular pairs in each direction. In particular each cluster has size $|V_i| \geq (1+\alpha /2)n/k$ . Form a reduced graph $R^\star$ with vertex set $[k]$ in which $i \to j$ is an edge if and only if the graph $G[V_i \to V_j]$ of edges directed from $V_i$ to $V_j$ is regular and dense. The reduced graph $R^\star$ then inherits an analogous minimum semidegree condition from $G$ , namely that $\delta ^0(R^\star ) \geq (1/2+\alpha /2)k$ . The full definitions and details for this step are presented in a more general form in Section 3.3.

2. (ii) The minimum semidegree of $R^\star$ is enough to ensure that $R^\star$ contains a spanning subgraph $R$ which is a $d$ -regular expander, meaning that every proper nonempty subset $S \subseteq V(R)$ has $|N^+(S)|,\, |N^-(S)| \gt |S|$ and every vertex $i \in V(R)$ has both indegree and outdegree of precisely $d$ . A more general version of this statement, along with a key mixing property of expander digraphs, is presented in Section 3.7.

3. (iii) Allocation: We now allocate the vertices of $T$ to clusters using a randomised allocation algorithm, which builds a homomorphism $\varphi \,:\, T \to R$ one vertex at a time. First, choose a root $r$ of $T$ and define $\varphi (r) \in [k]$ arbitrarily. Next iterate the following step: choose a vertex $x \in V(T)$ for which $\varphi (x)$ has been defined, say $\varphi (x) = u_x$ , but for which $\varphi (y)$ has not been defined for any child $y$ of $x$ . Choose vertices $u^+_x \in N^+_R(u_x)$ and $u^-_x \in N^-_R(u_x)$ uniformly at random, and set $\varphi (y) = u^+_x$ for every child $y$ of $x$ which is an outneighbour of $x$ , and $\varphi (y) = u^-_x$ for every child $y$ of $x$ which is an inneighbour of $x$ . Proceed in this manner until $\varphi (x)$ is defined for every vertex of $T$ , and observe that the resulting map $\varphi$ is then a homomorphism $\varphi : T \to R$ . Moreover, and crucially, so long as the maximum degree $\Delta (T)$ of $T$ is not too large, the mixing property of expander digraphs ensures that with high probability $\varphi$ allocates the vertices of $T$ approximately uniformly to the clusters of $R$ , with, say, for each $i \in [k]$ at most $(1+\alpha /3)n/k$ vertices $x \in V(T)$ having $\varphi (x) = i$ . The proof of this statement, for a more general version of this algorithm, is presented in Section 3.8.

4. (iv) Embedding: Finally we form a copy of $T$ in $G$ by greedily embedding each vertex $x$ of $T$ within the cluster $V_{\varphi (x)}$ of $G$ to which it was allocated, starting with the root $r$ . Each time we embed a vertex $x$ in a cluster $V_{u_x}$ we reserve sets of size $\textrm {O}(\sqrt {n})$ in the clusters $V_{u^-_x}$ and $V_{u^+_x}$ in which the children of $x$ which are respectively inneighbours and outneighbours of $x$ will be embedded; no other vertices may be embedded within these sets until all children of $x$ have been embedded. In this way we avoid treading on our toes’ by occupying all of the inneighbours or outneighbours of $x$ whilst they are still required for children of $x$ ; moreover the fact that edges of $R$ correspond to dense regular pairs in $G$ is sufficient for us to successfully choose appropriate embeddings within the specified sets. For this process to succeed we need the order in which we proceed through the vertices of $T$ to have the property that at any time there are not too many vertices which have been embedded but have a child vertex yet to be embedded; the notion of a tidy ancestral order presented in Section 3.1 captures what we need for this, whilst the way in which we choose the reserved sets is explained in Section 3.4.

A bound on $\Delta (T)$ is crucial for this argument to succeed, since we need the randomised allocation algorithm to distribute the vertices of $T$ approximately uniformly among the $k$ clusters (by contrast, the vertices of a star would be allocated in a highly-unbalanced way). To achieve this, we specifically need the distance between almost all pairs of vertices of $T$ to be significantly larger than $k$ , and indeed the bound on $\Delta (T)$ given in Theorem 4 is chosen to achieve precisely this property (see Sections 3.7 and 3.8 for more details).

2.2. Sketch proof for spanning trees

Let us now consider the case of Theorem 4(2) in which $Q$ is a tree on $n$ vertices and $G$ is a directed graph on $n$ vertices with $\delta ^0(G) \geq (1/2 +\alpha )n$ . Again, let $T$ be an orientation of $Q$ . The difference with the previous case is that our aim is now a spanning embedding of $T$ in $G$ ; there is no ‘room to spare’. However, Theorem 4(2) does provide a linear number of either bare paths on seven vertices or edges incident to leaves; for the sake of this discussion we assume the former (the arguments for the latter are quite similar and somewhat simpler).

We begin by splitting $T$ into two not-too-small subtrees $T_1$ and $T_2$ with one vertex $v$ in common, which we take as the root of both trees. Without loss of generality $T_1$ contains a set $\mathcal{P}$ of bare paths on seven vertices such that $\mathcal{P}$ has small linear size; and moreover we may insist that all paths in $\mathcal{P}$ have the same ‘pattern’ (the sequence of directions of edges along the path). As before we apply the regularity lemma to obtain clusters $V_1, \ldots , V_k$ of equal size and a small set $V_0$ of bad vertices, and similarly as there we may define a reduced graph $R^\star$ with vertex set $[k]$ whose vertices correspond to clusters, and also obtain a subgraph $R$ of $R^\star$ on vertex set $[k]$ which is a $d$ -regular expander. However, we now additionally insist that $R$ contains a Hamilton cycle $H$ (this is possible by the minimum degree condition that $R$ inherits from $G$ ); without loss of generality we assume that $H$ has edges $1 \to 2 \to 3 \to {\cdots} \to 1$ . We also insist that for each edge $i \to j$ of $H$ the pair $G[i \to j]$ is dense and superregular; this can be achieved by deleting a small number of vertices from each cluster and adding them to $V_0$ .

Next we apply a modified version of the allocation algorithm presented above to $T_1$ : the change is that if $x$ is a non-initial vertex of a bare path in $\mathcal{P}$ , and $y$ is the parent of $x$ , then we instead set $\varphi (x)$ to be $\varphi (y)+1$ if $y \in N^+(x)$ and $\varphi (y)-1$ if $y \in N^-(x)$ . The effect is that internal edges of bare paths in $\mathcal{P}$ are allocated deterministically within the cycle $H$ rather than randomly within the regular expander $R$ . Despite this change, the allocation algorithm still allocates the vertices of $T_1$ approximately uniformly across $[k]$ with high probability; moreover, the initial vertices of bare paths in $\mathcal{P}$ are also allocated approximately uniformly across $[k]$ with high probability.

We now choose pairwise disjoint sets ${\mathcal{P}}^H, {\mathcal{P}}^0, {\mathcal{P}}^\diamond \subseteq {\mathcal{P}}$ such that $|{\mathcal{P}}^0| = |V^0|$ , whilst ${\mathcal{P}}^H$ and ${\mathcal{P}}^\diamond$ each have small linear size. Each of these collections of bare paths will play a distinct role in completing the embedding of $T$ in $G$ , as follows.

1. (i) We reallocate the internal vertices of paths $P \in {\mathcal{P}}^0$ in such a way that the middle vertex of each path $P$ can be embedded to a corresponding vertex of $V_0$ . Note that, since in the allocation step we are forming a homomorphism $\varphi : V(T) \to R^\star$ , for notational convenience we actually also include each bad vertex in $V_0$ as a vertex of $R^\star$ , so $V(R^\star ) = [k] \cup V_0$ , and for each vertex $v \in V_0$ our allocation will have $\varphi (x) = v$ for precisely one vertex $x$ , which will then be embedded to $v$ at the embedding step ( $x$ is the middle vertex of a path $P \in {\mathcal{P}}^0$ ). In this way we ensure that all vertices of $V_0$ are covered by our embedding of $T$ .

2. (ii) We reallocate the internal vertices of paths $P \in {\mathcal{P}}^\diamond$ to structures in $R$ called ‘diamonds’ the purpose of which is that for each such path $P$ we have two choices for the allocation of the middle vertex $v_{4}^{P}$ of $P$ , where each choice is consistent with the allocation of the neighbours $v_3^P$ and $v_5^P$ of $v_{4}^{P}$ in $P$ . This gives us the flexibility to reallocate the middle vertex $v_{4}^{P}$ of each path $P \in {\mathcal{P}}^\diamond$ after all other vertices have been allocated, so as to slightly adjust the number of vertices allocated to each cluster. (Details of diamonds and this reallocation process can be found in Section 3.6.)

3. (iii) Finally, we leave the allocation of paths in ${\mathcal{P}}^H$ unchanged; the internal vertices of these paths will be the vertices we embed at the very end to ‘finish off’ our spanning embedding of $T$ in $G$ .

Unfortunately, it is unavoidable that the reallocation of vertices in paths in ${\mathcal{P}}^0$ and ${\mathcal{P}}^\diamond$ may create greater imbalances in the allocation of vertices of $T_1$ to clusters than can be corrected by reallocating the middle vertices of paths in ${\mathcal{P}}^\diamond$ . This is the purpose of the tree $T_2$ : after we have applied the randomised allocation algorithm to $T_1$ , and subsequently reallocated vertices of paths in ${\mathcal{P}}^0$ and ${\mathcal{P}}^\diamond$ , we now apply a biased version of the allocation algorithm to $T_2$ , which with high probability allocates slightly more vertices to the clusters which were underoccupied by $T_1$ , and slightly fewer vertices to the clusters which were overoccupied by $T_1$ . The end result is that the overall allocation of vertices of $T$ to clusters is very close to uniform; specifically it is close enough that it can be made uniform by reallocating the middle vertices of paths in ${\mathcal{P}}^\diamond$ . That is, after this reallocation exactly the same number of vertices are allocated to each cluster, meaning that the allocation ‘fits’ perfectly.

Finally, we embed the vertices of $T$ in $G$ by the same embedding approach as described previously, with two modifications. Firstly, we take special care with the embeddings of paths $P \in {\mathcal{P}}^0$ , whose middle vertices $v_{4}^{P}$ must be embedded to the vertex $v \in V_0$ to which they were allocated; to do this we embed the remaining vertices of $P$ so that the neighbours $v_3^P$ and $v_5^P$ of $v_{4}^{P}$ are embedded to inneighbours/outneighbours (as appropriate) of the vertices in $V_0$ . Secondly, we do not embed the internal vertices of paths $P \in {\mathcal{P}}^H$ at this stage, but leave these until the very end. This ensures that there is a little ‘room to spare’ when embedding all vertices of $T$ other than those in paths in ${\mathcal{P}}^H$ , so the previous approach of reserving sets for the children of each vertex as it is embedded is still valid. Finally, to complete the embedding we need to embed the internal vertices of the paths in ${\mathcal{P}}^H$ to the vertices still unoccupied within each cluster. Since each path in ${\mathcal{P}}^H$ is allocated along a sequence of superregular pairs (corresponding to edges of $H$ ), this can be done by applying the four-layer theorem of Komlós, Sárközy, and Szemerédi (see Section 3.3).

2.3. Sketch of the full proof

For the full version of the lemma, the graph $Q$ that we seek to embed may not be a tree. However the conditions on $Q$ do ensure that we can find a small ‘ground set’ $V_{\mathrm{ground}} \subseteq V(Q)$ so that $V_{\mathrm{ground}}$ contains all vertices of $Q_0$ as well as all short paths between vertices of $Q_0$ . In particular, $Q {\smallsetminus } V_{\mathrm{ground}}$ is a forest $F$ . We embed the vertices of $V_{\mathrm{ground}}$ greedily in $G$ , and then proceed similarly as in the previous case, dividing our forest $F$ into two subforests $F_1$ and $F_2$ and allocating and embedding these similarly as we described for $T_1$ and $T_2$ . Some additional difficulties arise here in doing this: for example, components of $F$ may contain two vertices with neighbours in $V_{\mathrm{ground}}$ , and whilst one can serve as the root of the component, the other (which we call a secondary attachment) must be handled carefully to ensure that it is allocated and embedded appropriately given the embedding of its neighbour in $V_{\mathrm{ground}}$ . However, these difficulties can be overcome through careful modification to the allocation and embedding procedures (in particular, this is the purpose of the set $Z$ of bad vertices in the general allocation algorithm in Section 3.8).

3.1. Trees and forests

Let $T$ be a tree or oriented tree. It is often helpful to nominate a vertex $r$ of $T$ as the root of $T$ ; to emphasise this fact we sometimes refer to $T$ as a rooted tree. If so, then every vertex $x$ other than $r$ has a unique parent; this is defined to be the (sole) neighbour $p$ of $x$ in the unique path in $T$ from $x$ to $r$ , and $x$ is said to be a child of $p$ . In the same way we call $w$ a descendant of $v$ , and $v$ an ancestor of $w$ , if the path from $r$ to $w$ contains $v$ . We denote the set of children of $v$ by $C(v)$ , and similarly write $C^-(v)$ for $N^-(v)\cap C(v)$ and $C^+(v)$ for $N^+(v)\cap C(v)$ .

Now let $F$ be a forest or oriented forest, so each component of $F$ is a tree or oriented tree. If we choose a root vertex for each component tree, then the definitions of parent, child, descendant, and ancestor extend naturally to forests: we say that $u$ is a child (respectively parent, descendant or ancestor) of $v$ in $F$ if $u$ and $v$ are in the same component $T$ of $F$ and $u$ is a child (respectively parent, descendant or ancestor) of $v$ in $T$ .

An ancestral order of the vertices of a rooted tree $T$ is an order of $V(T)$ in which every non-root vertex appears later than its parent; observe that this implies that the roof of $T$ appears first in the order, and that each vertex appears prior to each of its descendants and later than each of its ancestors. In exactly the same way, given a forest $F$ and a root vertex for each component tree in $F$ , an ancestral order of the vertices of $F$ is an order of $V(F)$ in which every non-root vertex appears later than its parent. Note that an ancestral order of the vertices of a tree (or forest) specifies the root(s) of the tree (or forest), since the root of each component must appear prior to every other vertex of that component. Consequently, when considering trees or forests equipped with ancestral orders we do not separately specify the roots of the tree or tree components.

We say that an ancestral order of a forest $F$ is tidy if for any initial segment $\mathcal{I}$ of the order, at most $\log _2 |V(F)|$ vertices in $\mathcal{I}$ have a child not in $\mathcal{I}$ . Such orders were considered by Kühn, Mycroft, and Osthus [Reference Kühn, Mycroft and Osthus16] for the purpose of embedding trees in tournaments; in particular they proved the following lemma for trees (the statement for forests follows immediately).

Our proof strategy at various stages requires us to divide a tree into subtrees, which we capture by the following notion.

Note that distinct trees in a tree-partition $\mathcal{P}$ share at most one vertex; moreover, if $\mathcal{P}$ contains at least 2 trees, then each tree has at least one vertex in common with some other tree in $\mathcal{P}$ . The following lemma shows that any tree admits a tree-partition into two subtrees which splits a given set of vertices somewhat evenly.

We will use the following straightforward corollary to obtain a small set of vertices which splits a forest into not-too-large parts.

The following proposition will be used at the very start of the proof of Theorem 4 to obtain the set $V_{\mathrm{ground}}$ ; the stated properties will simplify the allocation and embedding process for the tree components that remain after this set is deleted. Note that in the proof we use the folklore result that every tree $T$ contains a vertex $x$ such that each component of $T - x$ has at most $|T|/2$ vertices; this can be proved by orienting each edge of $T$ in the direction that leads to the most vertices (choosing arbitrarily if tied) and taking $x$ to be a sink of the resulting orientation (which must exist since $T$ contains no cycle).

Our final result on trees is a simple proposition which enables us to assume without loss of generality that the oriented graph $Q$ that we wish to embed has at most one component which is a tree.

3.2. Estimates and bounds

We write $\mathbb{E}(X)$ for the expectation of a random variable $X$ , and write $\mathbb{P}(A)$ for the probability of an event $A$ . We use the following well-known Chernoff-type bounds.

We also use a concentration result of McDiarmid [Reference McDiarmid17], in a form stated by Sudakov and Vondrák [Reference Sudakov and Vondrák20].

Lemma 15. [Reference McDiarmid17, Reference Sudakov and Vondrák20] Fix $n\in \mathbb{N}$ and let $X_1,\ldots ,X_n$ be random variables taking values in $[0,1]$ such that $\mathbb{E}(\,X_i\mid X_1,\ldots ,X_{i-1}\,)\leq a_i$ for each $i\in [n]$ . If $\mu \geq \sum _{i=1}^n a_i$ , then for every $\delta$ with $0\lt \delta \lt 1$ we have

\begin{equation*} \mathbb{P}\Biggl (\,\sum _{i=1}^n X_i \gt (1+\delta )\mu \,\Biggr )\leq \mathrm{e}^{-\delta ^2\mu /3}. \end{equation*}

3.3. Regularity

As well as the strict notion of regularity – that $G$ is $r$ -regular if every vertex in $G$ has degree precisely $r$ – we will also work with an approximate notion of regularity for bipartite graphs, where a graph is ‘regular’ if its edges are ‘random-like’ in the sense that they are distributed roughly uniformly. More formally, let $G$ be a bipartite graph with vertex classes $A$ and $B$ . For any sets $X \subseteq A$ and $Y \subseteq B$ , we write $G[X,Y]$ for the bipartite subgraph of $G$ with vertex classes $X$ and $Y$ whose edges are the edges of $G$ with one endvertex in each of the sets $X$ and $Y,$ and define the density $d_G(X,Y)$ of edges between $X$ and $Y$ to be

\begin{equation*}d_G(X,Y) \,:\!=\, \frac {e\bigl (G[X,Y]\bigr )}{|X||Y|}.\end{equation*}

Let $d, \varepsilon \gt 0$ . We say that $G$ is $(d,\varepsilon )$ -regular if for all $X\subseteq A$ and all $Y\subseteq B$ such that $|X|\geq \varepsilon |A|$ and $|Y|\geq \varepsilon |B|$ we have $d_G(X,Y) = d \pm \varepsilon$ . (To avoid confusion, note that the notion of being $(d, \varepsilon )$ -regular will always have two parameters, whereas the notion of an $r$ -regular graph or digraph has only one parameter.) The following well-known result is immediate from this definition.

We say that a bipartite graph $G$ with vertex classes $A$ and $B$ is $({{d_\geq }}, \varepsilon )$ -regular if $G$ is $(d'\!, \varepsilon )$ -regular for some $d' \geq d$ . For small $\varepsilon$ , if $G$ is $(d, \varepsilon )$ -regular then almost all vertices of $A$ have degree close to $d|B|$ in $B$ and vice-versa. We say that $G$ is ‘superregular’ if no vertex has degree much lower than this. More precisely, $G$ is $(d,\varepsilon )$ -superregular if $G$ is $({{d_\geq }},\varepsilon )$ -regular and also for every $a\in A$ and $b\in B$ we have $\textrm {deg}(a,B) \geq (d-\varepsilon )|B|$ and $\textrm {deg}(b,A) \geq (d-\varepsilon )|A|$ . The analogous statement to Lemma  16 for superregular graphs does not hold in general, since a subset of one vertex class may entirely avoid the neighbourhood of a vertex in the other vertex class. However, we can give a similar statement for uniformly-random subsets of each vertex class.

The final step in the proof of Theorem 4 is to embed the last few vertices of $Q$ into the equally-few remaining unused vertices of the host graph $G$ . In the case where $Q$ has many pairwise disjoint edges incident to leaves, we will achieve this by using the following well-known corollary of Hall’s marriage theorem, that every balanced superregular bipartite graph contains a perfect matching.

In the case where $Q$ has many pairwise vertex-disjoint edges incident to leaves, we instead use the following lemma of Komlós, Sárközy, and Szemerédi [Reference Komlós, Sárközy and Szemerédi13] (they actually stated the lemma only for $\ell =4$ , but the statement for larger $\ell$ follows immediately by combining this with Lemma  18). For this, say that a graph $G$ is an $\ell$ -layer $(d,\varepsilon )$ -superregular graph if $V(G)=\bigcup _{i\in [\ell ]} V_i$ , where $V_1, \ldots , V_\ell$ are pairwise disjoint sets of equal size $\ell$ and $G[V_i,V_{i+1}]$ is $(d,\varepsilon )$ -superregular for each $i\in [\ell -1]$ .

Let $X$ and $Y$ be disjoint sets of vertices in a digraph $G$ . Observe that the underlying graph of $G[X \rightarrow Y]$ is then a bipartite graph with vertex classes $X$ and $Y$ . We say that $G[X \rightarrow Y]$ is $(d, \varepsilon )$ -regular (respectively $({{d_\geq }}, \varepsilon )$ -regular or $(d, \varepsilon )$ -superregular) to mean that this underlying graph is $(d, \varepsilon )$ -regular (respectively $({{d_\geq }}, \varepsilon )$ -regular or $(d, \varepsilon )$ -superregular). In this way we may apply the previous results of this subsection to digraphs.

The celebrated Regularity Lemma of Szemerédi [Reference Szemerédi21, Reference Szemerédi22] states that every sufficiently large graph admits a partition such that almost all pairs of parts are regular in the sense we discuss here. Alon and Shapira [Reference Alon, Duke, Lefmann, Rödl and Yuster1] gave the following analogous result for directed graphs.

We refer to the sets $V_1,\ldots ,V_k$ as the clusters of $G$ . For $d\in [0,1]$ , the reduced graph $R$ with parameters $\varepsilon ,\,d$ and $K'$ of $G$ is a digraph we obtain by applying Lemma  20 to $G$ with parameters $\varepsilon , d$ and $K'$ ; the digraph $R$ has vertex set $[k]$ and edges $i{{}\to {}} j$ precisely when $G'[V_i{{}\to {}} V_j]$ has density at least $d$ . The following lemma will be used to obtain the reduced graph required in the allocation stage of the proof of Theorem 4.

3.4. Homomorphisms

As described in Section 2, a key idea for the proof of Theorem 4 is to find an allocation of the vertices of a subforest $F$ of $Q$ to the clusters of a reduced graph $R$ of the host graph $G$ . This allocation is a homomorphism from $F \to R$ with properties that will enable us to later embed each vertex of $F$ into the cluster to which it is embedded. In preparation for this, we make the following definitions.

Let $H$ and $G$ be digraphs. A homomorphism $\varphi \,:\, H\to G$ is an edge-preserving map from $V(H)$ to $V(G)$ , meaning that every edge $u{{}\to {}} v\in E(H)$ is mapped to an edge $\varphi (u){{}\to {}} \varphi (v)\in E(G)$ . The $\varphi$ -indegree $\textrm {deg}_\varphi ^-(v)$ of $v\in V(H)$ in $G$ is $\bigl |\varphi \bigl (N_H^-(v)\bigr )\bigr |$ ; the $\varphi$ -outdegree $\textrm {deg}_\varphi ^+(v)$ of $v$ is defined similarly, and the $\varphi$ -degree of $v$ is $\textrm {deg}_\varphi (v)\,:\!=\, \textrm {deg}_\varphi ^-(v) + \textrm {deg}_\varphi ^+(v)$ . Note that vertices in $\varphi \bigl (N_H^-(v)\bigr )\cap \varphi (N_H^+(v)\bigr )$ are counted twice towards $\textrm {deg}_\varphi (v)$ . The maximum degree of $\varphi$ is $\Delta (\varphi )\,:\!=\,\max _{v\in V(H)} \textrm {deg}_\varphi (v)$ .

The embedding algorithm used in the proof of Theorem 4 relies on a property of large dense digraphs stated in Lemma  23. The form in which it is stated here is a generalisation of that used by Kühn, Mycroft, and Osthus [Reference Kühn, Mycroft and Osthus16, Lemma 2.5]. We begin with a definition.

Here is some motivation for Definition  22. We will embed the vertices of the tree one by one, and, after embedding a vertex $x\in V(T)$ , we reserve sets of vertices for the children of $x$ . If the reserved sets are always good, then this greedy embedding strategy will succeed. The next lemma states one sufficient condition for this to occur.

Proof. Introduce new constants $\varepsilon '$ and $d'$ with $\varepsilon \ll \varepsilon ' \ll \gamma \ll d' \ll {\frac {1}{q}},\, \beta ,\,d$ . By removing leaves if necessary, we may assume that $\varphi (x)\neq \varphi (y)$ whenever $x,y$ are both inleaves or both outleaves of $S$ . Indeed, this does not change the statement we are trying to prove, since the definition of $({\mathcal{U}}^+, {\mathcal{U}}^-,\beta ,\gamma ,\varphi ,m)$ -good is not affected by whether the number of in-leaves (respectively out-leaves) of $S$ mapped to a fixed $x\in V(R)$ is precisely one or another positive integer. So we may proceed assuming that $S$ has precisely $q$ leaves $x_1,\ldots ,x_q$ . For each $i \in [q]$ let $U_i \,:\!=\, U_{\varphi (x_i)}^\bullet$ , where $\bullet \in \{+, -\}$ is such that $x_i \in N_S^\bullet (c)$ . For each $i\in [q]$ , let $S_i\,:\!=\, S\bigl [\{c,x_1,\ldots ,x_i\}\bigr ]$ , so $S_q=S$ .

Fix $Y\subseteq V_{\varphi (c)}$ with $|Y| \geq \gamma m/2$ . For each $t\in [q]$ , each $t$ -tuple ${\mathbf{t}}=(v_1,\ldots ,v_t)$ of distinct vertices with $v_j\in U_j$ for each $j\in [t]$ , and each subset $Z \subseteq Y$ , we write $N^{S_t}({\mathbf{t}},Z)$ for the set of vertices $v\in Z$ such that mapping $c \mapsto v$ and $x_j\mapsto v_j$ for each $j\in [t]$ gives a homomorphism from $S_t$ to $G$ . Also let $T$ be the set of tuples $(v_1,\ldots ,v_q)$ of distinct vertices with $v_i\in U_i$ for each $i\in [q]$ ; call ${\mathbf{t}}\in T$ bad if $\bigl |N^S({\mathbf{t}},Y)\bigr |\lt d'|Y|$ and good otherwise.

Let $B$ be the set of all bad tuples in $T$ ; our first goal is to show that $B$ is small. To do this, set $V^0 \,:\!=\, Y$ , and suppose that we choose $v_i \in U_i$ for each $i \in [q]$ in turn, at each step setting $V^i\,:\!=\, N^{S_i} \bigl ((v_1,\ldots ,v_i),V^{i-1}\bigr )$ . This yields a tuple ${{\mathbf{t}}} = (v_1,\ldots ,v_q) \in T$ . For each $i \in [q]$ say that the choice of $v_i$ is bad if $|V^i| \lt (d/2)|V^{i-1}|$ . So if we do not make any bad choices, then for each $i \in [q]$ we have $|V^i| \geq (d/2)|V^{i-1}|$ , and in particular $|V^q| \geq (d/2)^q|V^0| \geq d'|Y|$ , so ${\mathbf{t}}$ is good. It follows that if the outcome $\mathbf{t}$ is not good, then for some $i \in [q]$ we must have made a bad choice. By definition of $V^i$ this implies that $\textrm {deg}^\circ (v_i,V^{i-1}) \lt |V^{i-1}|d/2$ , where $\circ \in \{+, -\}$ is such that $c\in N_S^\circ (x_i)$ . Now consider the smallest $i$ for which we made a bad choice of $v_i$ ; this minimality property implies that $|V^{i-1}| \geq (d/2)^{i-1}|V^{0}| \geq (\gamma d^{i-1}/2^{i}) |V_{\varphi (c)}|$ , and so $G[U_i \rightarrow V^{i-1}]$ (if $\circ = +$ ) or $G[V^{i-1} \rightarrow U_i]$ (if $\circ = -$ ) is $(d_\geq , \varepsilon ')$ -regular by Lemma  16. However, since $d/2 \lt d-\varepsilon '$ , it follows that at most $\varepsilon '|U_i| \leq \varepsilon ' m$ vertices $b_i\in U_i$ have $\textrm {deg}^\circ (b_i,V^{i-1}) \lt |V^{i-1}|d/2$ , so there are at most $\varepsilon ' m$ bad choices for $v_i$ . We conclude that if ${\mathbf{t}}$ is bad then we must have made the first bad choice when choosing $v_i$ for some $i \in [k]$ , at which point there were at most $\varepsilon ' m$ options for the bad choice, so in total we have $|B| \leq k\varepsilon ' m^q$ (since we have $m$ options for each other choice).

Now choose a set $X\subseteq Y$ of size $|X|= \sqrt {m}$ uniformly at random. For each ${\mathbf{t}}\in T$ which is good, $\bigl |{N^{S}({\mathbf{t}},X)}\bigr |$ has hypergeometric distribution with expectation $\bigl |{N^{S}({\mathbf{t}},Y)}\bigr | (|X|/|Y|) \geq d'|X| = d' \sqrt {m}$ . So by Theorem 14 the probability that $\bigl |{N^{S}({\mathbf{t}},X)}\bigr | \lt d'\sqrt {m}/2$ decreases exponentially with $m$ . Since $|T{\smallsetminus } B| \leq |T| \leq (\beta m)^q$ , by taking a union bound we conclude that with high probability every ${\mathbf{t}}\in T{\smallsetminus } B$ has $\bigl |{N^{S}({\mathbf{t}},X)}\bigr |\ge d' \sqrt {m}/2$ . Fix a choice of $X$ for which this event occurs.

It remains to show that $X$ is $({\mathcal{U}}^+, {\mathcal{U}}^-,\beta ,\gamma ,\varphi ,m)$ -good for $S$ . So fix subsets $W_j^\circ \subseteq U_j^\circ$ of size $|W_j^\circ | \geq \beta m$ for each $\circ \in \{+, -\}$ and each $j \in J^\circ$ . Let $T'$ be the set of tuples $(v_1,\ldots ,v_q)$ of distinct vertices with $v_i\in W_{\varphi (x_i)}^-$ for each $i\in [q]$ with $x_i \in N^-_S(c)$ and $v_i\in W_{\varphi (x_i)}^+$ for each $i\in [q]$ with $x_i \in N^-_S(c)$ . So $|T'|\ge (\beta m)^q - qm^{q-1}$ , and therefore at least $|T'| - |B| \geq (\beta m)^q - q^2m^{q-1} - k\varepsilon ' m^q \geq \beta ^q m^q/2$ tuples ${\mathbf{t}}\in T'$ are good and so have $\bigl |{N^{S}({\mathbf{t}},X)}\bigr |\geq d'\sqrt {m}/2$ . Let ${\mathcal{P}}\,:\!=\, \bigl \{\, (v,{\mathbf{t}}) : {\mathbf{t}}\in T',\, v\in {N^{S}({\mathbf{t}},X)}\,\bigr \}$ , so

(1) \begin{equation} |{\mathcal{P}}| \ge \frac {\beta ^q m^q}{2} \cdot \frac {d' \sqrt {m}}{2} \ge \frac {d'\beta ^q}{4} m^{q+1/2}. \end{equation}

In particular, at least $\gamma \sqrt {m}$ vertices $v^\star \in X$ must have $v^* \in {N^{S}({\mathbf{t}},X)}$ for at least $\gamma m^q$ tuples ${\mathbf{t}}\in T'$ . Indeed, we would otherwise contradict (1), since we would then have

\begin{equation*} |{\mathcal{P}}| \lt |T'| \cdot \gamma \sqrt {m} + |X| \cdot \gamma m^q \leq m^q \cdot \gamma \sqrt {m} + \sqrt {m} \cdot \gamma m^q \leq 2\gamma m^{q+1/2}. \end{equation*}

Each such $v^\star \in X$ must then have at least $\gamma m^q/m^{q-1} = \gamma m$ inneighbours in $W_j^-$ for each $j \in J^-$ and at least $\gamma m$ outneighbours in $W_j^+$ for each $j \in J^+$ , as required.

3.5. Matching vertices

We use two simple results about matchings, whose straightforward proofs we omit.