In this section we use the regularity method to prove Lemma 3.14, which states that Theorem 1.1 holds in the case whereT∆is small andG is almost-regular. Specifically, we consider directed treesT for which T∆is substantially smaller than the size of a cluster obtained by applying the regularity lemma to a tournament G; our approach here is essentially to select an appropriate
cluster or pair of clusters ofG in which to embed T∆so that we may then embed the components of T − T∆ in the remaining clusters ofG. We finish the section by combining Lemmas 3.14 and 2.32 to obtain Lemma 3.19, which states that Theorem 1.1 holds whenever G is almost-
regular.
To prove Lemma 3.14 we begin by applying the regularity lemma to partition the vertices of
G into clusters V1, . . . , Vk. We deduce from the fact thatG is almost-regular that every cluster
Vi has roughly the same number of edges entering it as leaving it. In Lemma 3.9 we consider the case where some cluster Vi has the property that for any other cluster Vj there are many edges from Vi to Vj and many edges from Vj to Vi. Here we show that may embed T in G by first embedding T∆ within the cluster Vi, and then using the regularity of edges between pairs of clusters to embed the remaining vertices ofT . In Lemma 3.13 we instead consider the
case where there is no such cluster. Here we show that we may embed T in G by the same
Lemma 3.14 holds in the case whereT∆has small outweight. In this case we select two clusters
ViandVj for which the corresponding vertices in the ‘reduced digraph’ ofG has small common outneighbourhood. We then embed the vertices ofT∆ within these two clusters, before using the regularity of edges between pairs of clusters to embed the remaining vertices ofT in G. So
in terms of the description of the regularity method given in the introduction, the subgraph we find in the reduced digraph is a single vertex or pair of vertices with given properties on the edges entering and leaving the corresponding cluster or clusters ofG.
We say that an oriented graph G on clusters V1, . . . , Vk of equal size is an ε-regular cluster
tournament if for anyi, j ∈ [k] with i 6= j the subdigraph G[Vi → Vj] is ε-regular and for any
i∈ [k] the subdigraph G[Vi] is a tournament. If G is a cluster tournament on clusters V1, . . . , Vk then we denote the density ofG[Vi → Vj] by dij for anyi, j ∈ [k] (the tournament G should be clear from the context). The following corollary of the digraph regularity lemma (Lemma 2.7) shows that any sufficiently large tournamentG contains an almost-spanning ε-regular cluster
tournamentG∗such that all vertices have similar in- and outdegrees in bothG and G∗.
Corollary 3.7 Suppose that1/n 1/M 1/M0 ε. LetGbe a tournament onnvertices.
Then there exist disjoint subsetsV1, . . . , Vk ⊆ V (G)of equal size and a subgraphG∗ ⊆ Gon
vertex setV1∪ · · · ∪ Vksuch that:
(i) M0 6k 6 M,
(ii) G∗is anε-regular cluster tournament,
(iii) Si∈[k]Vi >(1− ε)n,
(iv) d+G∗(x) > d
+
G(x)− 2εnfor all verticesx∈ V (G), and
Proof. Apply Lemma 2.7 withd = 0 to obtain a partition V0, . . . , VkofV (G) and a subgraph
G0 ⊆ G which satisfy the conditions of Lemma 2.7. In particular (i) and (iii) are satisfied. Now formG∗ fromG0[V
1∪ · · · ∪ Vk] by adding every edge of G for which both endvertices lie in the same clusterVi. SoG∗ ⊆ G, and by (7) of Lemma 2.7 and the fact that G∗[Vi] is a tournament for eachi∈ [k] we have (ii). Finally note that using (4) of Lemma 2.7 we have
d+G∗(x) > d+G0(x)− |V0| > d
+
G(x)− 2εn.
Similarlyd−G∗(x) > d−G(x)− 2εn using (5) of Lemma 2.7.
It follows immediately from the definition of regularity that ifU and V are sets of size m, and G[U → V ] is ε-regular with density d, then all but at most 2εm vertices of U have (d ± ε)m
outneighbours inV . The next lemma is a generalisation of this fact, considering the number of
outneighbours of vertices in one cluster within a cluster tournament.
Lemma 3.8 Suppose that 1/m 1/k ε ε0 1. LetG be anε-regular cluster tour-
nament on clusters V1, . . . , Vk, each of size m. Let Vj0 ⊆ Vj for each j ∈ [k]be fixed. Then
for any i, all but at most ε0m vertices of Vi have Pj∈[k]\{i}dij|Vj0| ± ε0km outneighbours in
S j∈[k]\{i}Vj0and P j∈[k]\{i}dji|Vj0| ± ε0kminneighbours in S j∈[k]\{i}Vj0.
Proof. Fix some i ∈ [k]. Then let L be the set of all j ∈ [k] \ {i} such that |V0
j| > εm and dij > √ε. For each j ∈ L, let Aj denote the set of vertices ofVi which have fewer than
(1−√ε)dij|Vj0| outneighbours in Vj0. Then for each j ∈ L, the subdigraph of G[Vi → Vj] induced by Aj and Vj0 has density less than (1−
√
ε)dij 6 dij − ε. Since G[Vi → Vj] is
ε-regular with density dij, and|Vj0| > εm, we must have |Aj| < εm.
Then |N+(v)∩[ j∈L Vj0| > X j∈L:v /∈Aj (1−√ε)dij|Vj0| > X j∈[k]\{i} (1−√ε)dij|Vj0| − X j∈[k]\(L∪{i}) dij|Vj0| − X j∈L:v∈Aj dij|Vj0| > X j∈[k]\{i} dij|Vj0| − √ εkm−√εkm−√ε|L|m > X j∈[k]\{i} dij|Vj0| − 3 √ εkm.
Since at most√εm vertices v∈ Viappear in more than√ε|L| of the sets Ajwithj ∈ L, we may conclude that there are at most√εm vertices v∈ Viwith fewer than
P
j∈[k]\{i}dij|Vj0|−3
√ εkm
outneighbours inSj∈[k]\{i}V0
j. A similar argument shows that there are at most
√
εm vertices v ∈ Vi with more thanPj∈[k]\{i}dij|Vj0| + 3
√
εkm outneighbours inSj∈[k]\{i}V0 j.
Now, letL0be the set of allj ∈ [k] such that |V0
j| > εm and dji >√ε. Then the same argument applied to inneighbours rather than outneighbours shows that there are at most√εm vertices v ∈ Vi with fewer than
P
j∈[k]\{i}dji|Vj0| − 3
√
εkm inneighbours inSj∈[k]\{i}Vj0 and at most
√
εm vertices v ∈ Viwith more than
P
j∈[k]\{i}dji|Vj0| + 3
√
εkm inneighbours inSj∈[k]\{i}Vj0.
Sinceε ε0, this completes the proof.
The next two lemmas are used in the proof of Lemma 3.14; we state them separately as we also refer to them in Section 3.5. Both of these consider an ε-regular cluster tournament G
onk clusters with the property that for some cluster Vi the density of edges leavingVi and the density of edges entering Vi are each roughly1/2. Lemma 3.9 considers the case where for many clusters Vj the density of edges between Vi andVj is large in both directions, showing that in this case G contains a copy of a directed tree T of the type considered. Lemma 3.13
andVj is small in one direction, showing that in this caseG contains a copy of T provided that
T∆has large inweight and large outweight.
Lemma 3.9 Suppose that 1/n 1/∆0, β 1/k ε γ α 1/∆ 1. Let T
be a directed tree on n vertices with |T∆0| 6 βn and |T∆| > 2, and let G be an ε-regular
cluster tournament on clustersV1, . . . , Vk, each of sizem > 2(1− γ)n/k. Suppose also that for
somei∈ [k]we have X j∈[k]\{i} dij > (1− 3γ)k 2 and X j∈[k]\{i} dji > (1− 3γ)k 2 ,
and also that there are at leastαk values ofj ∈ [k] \ {i}such thatdij > αanddji > α. Then
Gcontains a copy ofT.
Proof. Fix such a value ofi, and introduce a new constant ε0 withε ε0 γ. Since ∆ 6 ∆0, we must haveT∆ ⊆ T∆0. Also, since|T∆| > 2, we may choose an edge t− → t+ofT∆, which
therefore is also an edge ofT∆0. LetT+andT−be the two components formed when this edge
is deleted fromT , labelled so that t+ ∈ T+andt− ∈ T−. Similarly, letT+
∆0 andT∆−0 be the two
components formed by the deletion of the edget− → t+fromT
∆0, labelled witht+ ∈ T∆+0 and
t− ∈ T∆−0. Then T+ andT− partition the vertices ofT , and there is precisely one edge of T
betweenT+andT−, which is directed towardsT+. Furthermore, sincet− → t+ was an edge ofT∆, by Proposition 2.19(ii) we have|T+|, |T−| > n/∆.
LetJ ⊆ [k] \ {i} satisfy |J| > αk and also that for any j ∈ J we have dij > α and dji > α. Then Pj∈Jdij > α2k andPj∈Jdji > α2k. By Lemma 3.8 (applied with Vj0 = ∅ for each
j /∈ J) at most ε0m vertices of V
i have fewer than
X
j∈J
dijm− ε0km > α2km− ε0km >
α2km
outneighbours inSj∈JVjor fewer than
P
j∈J djim−ε0km > α2km/2 inneighbours in
S
j∈JVj. Also by Lemma 3.8 at mostε0m vertices of V
i have fewer than
X
j∈[k]\{i}
dijm− ε0km >
(1− 3γ − 2ε0)km
2 >(1− 5γ)n (3.11)
outneighbours inSj∈[k]\{i}Vj or fewer thanPj∈[k]\{i}djim− ε0km > (1− 5γ)n inneighbours inSj∈[k]\{i}Vj. Finally, at mostm/2 + 1 vertices of Vihave fewer thanm/4 inneighbours in Vi. So we may choose a setS+ofm/10 vertices of V
iwhich do not fall into any of these categories. Since|T∆+0| 6 |T∆0| 6 βn 6 m/30, by Theorem 1.2 we may embed T
+
∆0 inS+. LetS
+
∆0 be the
set of vertices ofS+ occupied by this embedding ofT+
∆0, and letv+ be the vertex to whicht+
was embedded. Recall that|T−| > n/∆, so
|T+| = n − |T−| 6 (1 − ∆1)n.
Furthermore, every component ofT+− T+
∆0 is a component of T − T∆0 and thus has order at
most n/∆0 by Proposition 2.19(iv). So by (3.10) and (3.11), and since γ 1/∆, we may apply Lemma 3.3(b) to extend the embedding ofT∆+0 inS
+
∆0 to an embedding ofT+ inS
+ ∆0 ∪
S
j∈[k]\{i}Vj so that at leastα2n/3 vertices of
S
j∈JVj are occupied by this embedding ofT+.
Now, at leastm/4− m/10 = 3m/20 vertices of Vi \ S∆+0 are inneighbours of v+. For each
j ∈ [k] \ {i}, let oj denote the number of vertices ofVj which are occupied by our embedding ofT+, and letVj0 ⊆ Vjconsist of those vertices ofVjwhich are not occupied by this embedding. So|Vj0| = m − oj for eachj. Note that since dij+ dji 61 we have dij 61− α for each j ∈ J.
Then by Lemma 3.8, at mostε0m vertices of Vi have fewer than X j∈[k]\{i} dij(m− oj)− ε0km > X j∈[k]\{i} dijm− ε0km− X j∈J dijoj − X j∈[k]\({i}∪J) dijoj (3.11) > (1− 5γ)n − (1 − α)X j∈J oj− X j∈[k]\({i}∪J) oj >(1− 5γ)n − X j∈[k]\{i} oj + α X j∈J oj >(1− 5γ)n − X j∈[k]\{i} oj + α3n/3 > n− X j∈[k]\{i} oj + 2n ∆0 (3.12) outneighbours inSj∈[k]\{i}V0 j or fewer than X j∈[k]\{i} dji(m− oj)− ε0km > n− X j∈[k]\{i} oj + 2n ∆0,
inneighbours in Sj∈[k]\{i}Vj0. So we may choose a setS− ofm/10 vertices of Vi \ S∆+0, none
of which fall into these two categories, and all of which are inneighbours ofv+. Since |T− ∆0| 6
|T∆0| 6 βn 6 m/30, by Theorem 1.2 we may embed T∆−0 inS−. LetS∆−0 be the set of vertices
ofS−occupied by this embedding ofT−
∆0. Then since
|T−| = n − |T+| 6 n − X j∈[k]\{i}
oj,
the right hand side of (3.12) is at least |T−| + 2n/∆0. Also every component of T− − T− ∆0
is a component of T − T∆0 (and so has order at most n/∆0 by Proposition 2.19(iv)). So
by Lemma 3.3(b) we may extend the embedding of T∆−0 in S∆−0 to an embedding of T− in
S∆−0 ∪
S
j∈[k]\{i}Vj0. Then the embeddings ofT+ andT− do not overlap, and so together these
Given anε-regular cluster tournament G on clusters V1, . . . , Vk, we define the reduced digraph
ofG with parameter d, denoted RG(d), to be the directed graph on vertex set [k] in which i→ j if and only ifdij > d. Observe that since dij + dji 61 for any i and j, if d > 1/2 then RG(d) is an oriented graph.
Lemma 3.13 Suppose that 1/n 1/∆0, β 1/k ε γ α 1. Let T be a
directed tree on n vertices with |T∆0| 6 βn, and let y and z be the outweight and inweight
ofT∆0 respectively. LetG be anε-regular cluster tournament on clustersV1, . . . , Vk, each of
sizem > 2(1− γ)n/k. Suppose that for somei∈ [k]we have
X j∈[k]\{i} dij > (1− 3γ)k 2 and X j∈[k]\{i} dji> (1− 3γ)k 2 ,
and also that there are at mostαk values ofj ∈ [k] \ {i}such thatdij >αanddji >α. Then:
(i) There are at most2αkvalues ofj ∈ [k] \ {i}such thatdij < 1− 2αanddji < 1− 2α.
(ii) LetR := RG(1− 2α). Then|NR+(i)|, |NR−(i)| > (1 − 10α)k/2.
(iii) Ify, z > 14αn, thenGcontains a copy ofT.
Proof. Fix such ani, and introduce a new constant ε0 withε ε0 γ. For (i), note that since
dij + dji61 for any j ∈ [k] \ {i}, and
X
j∈[k]\{i}
(dij + dji) > (1− 3γ)k,
there are at most3√γk 6 αk values of j ∈ [k] \ {i} for which dij + dji < 1−√γ. So there are at most2αk values of j ∈ [k] \ {i} for which dij < 1− α −√γ and dji < 1− α −√γ, so (i) holds.
For (ii), observe that by (i) we have (1− 3γ)k 2 6 X j∈[k]\{i} dij 6 X j∈[k]\{i} dij>1−2α dij + X j∈[k]\{i} dij,dji<1−2α dij+ X j∈[k]\{i} dij62α dij 6|NR+(i)| + 2αk + 2αk,
so|NR+(i)| > (1 − 10α)k/2. A similar calculation shows that |N−
R(i)| > (1 − 10α)k/2.
For (iii), let N+ and N− denoteNR+(i) and NR−(i) respectively, and let V+ := Sj∈N+Vj and
V− :=S
j∈N−Vj, soV
+andV− are disjoint. By Lemma 3.8,V
i contains at mostε0m vertices with fewer than
X j∈N+ dijm− ε0km >|NR+(i)|(1 − 2α)m − ε0km > (1− 10α)(1 − 2α)km 2 − ε 0km > (1− 12α − 2ε 0)km 2 >(1− 13α)n
outneighbours in V+ and at most ε0m vertices with fewer thanP
j∈N−djim− ε0km > (1−
13α)n inneighbours in V−. Choose a set S of m/2 vertices of V
i, not including any of these at most2ε0m vertices. Since |T∆0| 6 βn 6 m/6, by Theorem 1.2 we may embed T∆0 inS.
LetS∆0 be the set of vertices ofS occupied by this embedding of T∆0. Also letT1 be the tree
formed byT∆0 and all of its outcomponents, and letT2 be the tree formed byT∆0 and all of its
incomponents. Note that all of these out- and incomponents have order at mostn/∆0 αn by Proposition 2.19(iv). In addition|T1| = n − z 6 (1 − 14α)n and |T2| = n − y 6 (1 − 14α)n. So by Lemma 3.3(c) we may extend the embedding of T∆0 in S∆0 to an embedding of T1 in
S∆0 ∪ V+. Similarly by Lemma 3.3(c) we may extend the embedding of T∆0 in S∆0 to an
embedding ofT2 inS∆0 ∪ V−. Then these embeddings do not overlap outsideT∆0, so we may
The next lemma shows that Sumner’s universal tournament conjecture holds whenever n is
sufficiently large,G is an almost-regular tournament and the core tree T∆ofT is small. Actually we prove a slightly stronger result in this case, considering a tournament on fewer than2n− 2
vertices. Later on we make use of the fact that we have a little room to spare in the order of the tournament. Much of the work for this lemma is done by the two previous lemmas.
Lemma 3.14 Suppose that1/n 1/∆0, β γ 1/∆ 1. LetT be a directed tree on n
vertices such that|T∆0| 6 βnand|T∆| > 2. LetGbe aγ-almost-regular tournament on at least
(2− γ)nvertices. ThenGcontains a copy ofT.
Proof. Introduce new constantsε, ε0, α, M, and M0 with
1/n 1/∆0, β 1/M 1/M0 ε ε0 γ α 1/∆ 1.
If|G| > (2 + γ)n, then G contains a copy of T by Theorem 1.5(1). So we may assume that
|G| = (2 ± γ)n. Observe that d+(v), d−(v) > (1− γ)(|G| − 1)/2 > (1 − 2γ)n for all v ∈ G. Since∆ 6 ∆0, we must haveT∆⊆ T∆0. Also, since|T∆| > 2, we may choose an edge t−→ t+
ofT∆, which must also lie in T∆0. Let T+ and T− be the two components formed when this
edge is deleted fromT , labelled so that t+∈ T+andt− ∈ T−. Similarly, letT+
∆0 andT∆−0 be the
two components formed by the deletion of the edget− → t+fromT
∆0, labelled witht+ ∈ T∆+0
andt− ∈ T−
∆0. ThenT+andT−partition the vertices ofT , and there is precisely one edge of T
betweenT+andT−, which is directed towardsT+. Furthermore,|T+|, |T−| > n/∆.
Let disjoint subsetsV1, . . . , Vkand a subgraphG∗ ⊆ G satisfy the conditions of Corollary 3.7. SoM0 6k 6 M, and G∗is anε-regular cluster tournament on clusters V1, . . . , Vkof equal size
m, where 2(1− γ)n