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arXiv:2207.09146v2 [math.CO] 17 Sep 2022
Forests and the Strong Erd¨os-Hajnal
Property
September 20, 2022
Soukaina ZAYAT 1
Abstract
An equivalent directed version of the celebrated unresolved conjecture of Erd¨os and
Hajnal proposed by Alon et al. states that for every tournament Hthere exists ǫ(H)>
0 such that every H-free n-vertex tournament Tcontains a transitive subtournament
of order at least nǫ(H). A tournament H has the strong EH-property if there exists
c > 0 such that for every H-free tournament Twith |T|>1, there exist disjoint vertex
subsets Aand B, each of cardinality at least c|T|and every vertex of Ais adjacent
to every vertex of B. Berger et al. proved that the unique five-vertex tournament
denoted by C5, where every vertex has two inneighbors and two outneighbors has the
strong EH-property. It is known that every tournament with the strong EH-property
also has the EH-property. In this paper we construct an infinite class of tournaments
−the so-called spiral galaxies and we prove that every spiral galaxy has the strong
EH-property.
Keywords. Tournament, ordering, pure pair, Erd¨os-Hajnal Conjecture, forest.
1 Introduction
Let Gbe an undirected graph. A clique in Gis a set of pairwise adjacent vertices and a
stable set in Gis a set of pairwise nonadjacent vertices. A tournament is an orientation of
a complete graph. A tournament is transitive if it contains no directed cycle. Let Hbe a
tournament. If (u, v)∈A(H), then we say that uis adjacent to v, and we write u→v.
In this case, we also say that vis adjacent from u, and we write v←u. For two disjoint
sets of vertices V1, V2of H, we say that V1is complete to V2(equivalently V2is complete
from V1) if every vertex of V1is adjacent to every vertex of V2. We say that a vertex vis
complete to (resp. from) a set Vif {v}is complete to (resp. from) V, and we write v→V
(resp. v←V). Let X⊆V(H). The subtournament of Hinduced by Xis denoted by
H|X. Let Sbe a tournament. We say that Hcontains Sif Sis isomorphic to H|Xfor
1Department of Mathematics, Lebanese University, Hadath, Lebanon. (soukaina.zayat.96@outlook.com)
1
some X⊆V(H). If Hdoes not contain S, we say that His S-free.
In 1989 Erd¨os and Hajnal proposed the following conjecture [7] (EHC):
Conjecture 1 For any undirected graph Hthere exists ǫ(H)>0such that every n-vertex
undirected graph that does not contain Has an induced subgraph contains a clique or a
stable set of size at least nǫ(H).
In 2001 Alon et al. proved [1] that Conjecture 1 has an equivalent directed version, as
follows:
Conjecture 2 For any tournament Hthere exists ǫ(H)>0such that every H-free tour-
nament with nvertices contains a transitive subtournament of order at least nǫ(H).
A tournament Hhas the Erd¨os-Hajnal property (EH-property) if there exists ǫ(H)>0 such
that every H-free tournament Twith nvertices contains a transitive subtournament of size
at least nǫ(H).
Let θ= (v1, ..., vn) be an ordering of the vertex set V(T) of an n-vertex tournament T.
We say that a vertex vjis between two vertices vi, vkunder θ= (v1, ..., vn) if i < j < k
or k < j < i. An arc (vi, vj) is a backward arc under θif i > j. The set of backward
arcs of Tunder θis denoted by AT(θ). The backward arc digraph of Tunder θ, denoted
by B(T, θ), is the subdigraph that has vertex set V(T) and arc set AT(θ). We say that
V(T)is the disjoint union of X1,...,Xtunder θif V(T) is the disjoint union of X1,...,Xt
and A(B(T, θ)) =
t
[
i=1
A(B(T|Xi, θi)), where θiis the restriction of θto Xi.
A tournament Son pvertices with V(S) = {u1, u2, ..., up}is a right star (resp. left
star) (resp. middle star) if there exists an ordering β= (u1, u2, ..., up) of its vertices such
that A(B(S, β)) = {(up, ui) : i= 1, ..., p −1}(resp. A(B(S, β)) = {(ui, u1) : i= 2, ..., p})
(resp. A(B(S, β)) = {(ui, um) : i=m+ 1, ..., p} ∪ {(um, ui) : i= 1, ..., m −1}, where
m∈ {2, ..., p −1}). In this case we write S={u1, u2, ..., up}and we call βaright star
ordering (resp. left star ordering) (resp. middle star ordering) of S,up(resp. u1) (resp.
um) the center of S, and u1, ..., up−1(resp. u2, ..., up)(resp. u1, ..., um−1, um+1, ..., up) the
leaves of S. A frontier star is a left star or a right star. A star is a middle star or a frontier
star. A star ordering is a right or left or middle star ordering.
Astar S={vi1, ..., vit}of Tunder θ(where i1< ... < it) is the subtournament of T
induced by {vi1, ..., vit}such that Sis a star and Shas the star ordering (vi1, ..., vit) under
θ(i.e (vi1, ..., vit) is the restriction of θto V(S) and (vi1, ..., vit) is a star ordering of S).
Let Tbe a tournament and assume that there exists an ordering θof its vertices such
that V(T) is the disjoint union of V(S1), ..., V (Sl), X under θ, where S1, ..., Slare the stars
of Tunder θ, and for every x∈X,{x}is a singleton component of B(T, θ). In this case T
is called nebula and θis called a nebula ordering. If all the stars of Tunder θare frontier
stars and no center of a star is between leaves of another star under θ, then θis called a
galaxy ordering and Tis called a galaxy under θ. If moreover Xis empty, then Tis called
aregular galaxy under θ.
2
In [2] Berger et al. proved that every galaxy has the EH-property, and in [5, 9, 10]
Conjecture 2 was proved for more general classes of tournaments.
Let Tbe a tournament. A pure pair in Tis an ordered pair (A, B) of disjoint subsets
of V(T) such that every vertex in Ais adjacent to every vertex in B, and its order denoted
by O(A, B) is min(|A|,|B|). Define P(T) := max{O(A, B) : (A, B) is a pure pair of T}.
We call T α-coherent for α > 0 if P(T)< α |T|.
A tournament Hhas the strong EH-property if there exists α > 0 such that for every H-
free tournament Twith |T|>1, we have: P(T)≥α|T|. It is easy to see that for every
tournament with the strong EH-property also has the EH-property [2, 4].
In [4] Berger et al. proved that the unique five-vertex tournament denoted by C5, where
every vertex has two inneighbors and two outneighbors has the strong EH-property. In
[6] Chudnovsky et al. asked if it might be true that a tournament Hhas the strong EH -
property if and only if its vertex set has an ordering in which the backward arc digraph of
Hunder this ordering is a forest. Chudnovsky et al. proved that the necessary condition is
true:
Theorem 1.1 [6] If a tournament has the strong EH-property then there is an ordering of
V(H)for which the backward arc digraph is a forest.
Unfortunately, the following is still open:
Conjecture 3 If a tournament Hhas an ordering of its vertices for which the backward
arc digraph of Hunder this ordering is a forest, then Hhas the strong EH-property.
In [6] Chudnovsky et al. made a small step towards Conjecture 3 by showing that if a
tournament His C5-free and it has an ordering θof its vertices such that |AH(θ)|≤ 3, then
it has the strong EH-property. In particular, they proved that every tournament with at
most six vertices has the property, except for three six-vertex tournaments that they could
not decide.
In [10] the author and Ghazal proved that every galaxy with spiders has the EH-property
(see [10] for the detailed description of galaxies with spiders).
In this paper we prove Conjecture 3 for an infinite family of tournaments −the so-
called spiral galaxies. Spiral galaxies are nebulas satisfying some conditions to be explained
in details in Section 3. Also note that every spiral galaxy is a galaxy with spiders.
2 Smooth (c, λ, w)-structure
Let Tbe a tournament and let X, Y ⊆V(T) be disjoint. Denote by eX,Y the number of
directed arcs (x, y), where x∈Xand y∈Y. The directed density from Xto Yis defined
as d(X, Y ) = eX,Y
|X|.|Y|.
3
Lemma 2.1 [2] Let A1, A2be two disjoint sets such that d(A1, A2)≥1−λand let 0<
η1, η2≤1. Let b
λ=λ
η1η2. Let X⊆A1, Y ⊆A2be such that |X| ≥ η1|A1|and |Y| ≥ η2|A2|.
Then d(X, Y )≥1−b
λ.
The following is introduced in [3].
Let c > 0,0< λ < 1 be constants, and let wbe a {0,1}-vector of length |w|. Let Tbe a
tournament with |T|=n. A sequence of disjoint subsets χ= (A1, A2, ..., A|w|) of V(T) is a
smooth (c, λ, w)-structure if:
•whenever wi= 0 we have |Ai| ≥ cn (we say that Aiis a linear set).
•whenever wi= 1 the tournament T|Aiis transitive and |Ai| ≥ c.tr(T) (we say that Aiis
atransitive set).
•d({v}, Aj)≥1−λfor v∈Aiand d(Ai,{v})≥1−λfor v∈Aj, i < j (we say that χis
smooth).
Theorem 2.2 [4] Let Sbe a f-vertex tournament, let wbe an all-zero vector. Then there
exist c > 0such that every S-free tournament contains a smooth (c, 1
f, w)-structure.
Let (S1, ..., S|w|) be a smooth (c, λ, w)-structure of a tournament T, let i∈ {1, ..., |w|}, and
let v∈Si. For j∈ {1,2, ..., |w|}\{i}, denote by Sj,v the set of the vertices of Sjadjacent
from vfor j > i and adjacent to vfor j < i.
Lemma 2.3 [9] Let 0< λ < 1,0< γ ≤1be constants and let wbe a {0,1}−vector. Let
(S1, ..., S|w|)be a smooth (c, λ, w)-structure of a tournament Tfor some c > 0. Let j∈
{1, ..., |w|}. Let S∗
j⊆Sjsuch that |S∗
j| ≥ γ|Sj|, and let A={x1, ..., xk} ⊆ [
i6=j
Sifor some
positive integer k. Then |\
x∈A
S∗
j,x| ≥ (1 −kλ
γ)|S∗
j|. In particular |\
x∈A
Sj,x| ≥ (1 −kλ)|Sj|.
Proof. The proof is by induction on k. without loss of generality assume that x1∈Siand
j < i. Since |S∗
j| ≥ γ|Sj|, then by Lemma 2.1, d(S∗
j,{x1})≥1−λ
γ. So 1−λ
γ≤d(S∗
j,{x1}) =
|S∗
j,x1|
|S∗
j|. Then |S∗
j,x1| ≥ (1 −λ
γ)|S∗
j|and so true for k= 1. Suppose the statement is true for
k−1.
|\
x∈A
S∗
j,x|=|(\
x∈A\{x1}
S∗
j,x)∩S∗
j,x1|=|\
x∈A\{x1}
S∗
j,x|+|S∗
j,x1| − |(\
x∈A\{x1}
S∗
j,x)∪S∗
j,x1| ≥ (1 −
(k−1)λ
γ)|S∗
j|+ (1 −λ
γ)|S∗
j| − |S∗
j|= (1 −kλ
γ)|S∗
j|.
3 Main Result
Let Hbe a tournament, let θ= (1, ..., t) be an ordering of its vertices, and let 2 ≤r≤t−5.
His called middle-pair-star under θ= (1, ..., t) if either AT(θ) = {(r+ 3, r),(r+ 4, r + 1)} ∪
{(r, i) : i= 1, ..., r} ∪ {(i, r + 4) : i=r+ 5, ..., t}or AT(θ) = {(r+ 3, r),(r+ 4, r + 1)} ∪ {(r+
4, i) : i= 1, ..., r} ∪ {(i, r) : i=r+ 5, ..., t}. In this case we call the vertices r, ..., r + 4 the
golden vertices of H,r, r+4 the centers of H, and 1, ..., r−1, r+5, ..., t the leaves of H.His a
left-pair-star under θif AH(θ) = {(4,1),(5,2)}∪{(i, 5) : i= 6, ..., j}∪{(i, 1) : i=j+1, ..., t}
4
or AH(θ) = {(4,1),(5,2)}∪{(i, 1) : i= 6, ..., j}∪{(i, 5) : i=j+1, ..., t}, where 6 ≤j≤t−1.
In this case we call the vertices 1, ..., 5 the golden vertices of H, 1,5 the centers of H, and
6, ..., t the leaves of H.Right-pair-stars are defined similarly. If His a middle-pair-star
(resp. right-pair-star) (resp. left-pair-star) under θ, then θis called star ordering of H.
Let Tbe a tournament drawn under an ordering θ= (v1, ..., vn) of its vertices. A pair-star
P:= {vi1, ..., vit}of Tunder θ(where i1< ... < it) is an induced subtournament of Twith
vertex set {vi1, ..., vit}, such that: (vi1, ..., vit) is its star ordering, the golden vertices are
consecutive under θ, and leaves incident to the same center are consecutive under θ.
A tournament His a path-galaxy under an ordering θof its vertices if His a galaxy under
θ, and the leaves of every star are consecutive under θ. A tournament His a regular path-
galaxy under θif His a regular galaxy under θwhich is a path-galaxy, and moreover all the
stars of Hunder θare of the same size. Let H be a tournament and let θbe an ordering
of its vertices such that V(H) is the disjoint union of V(S1), ...,V(Sl), X under θ, where
S1, ..., Slare the pair-stars of Hunder θ,H|Xis a path-galaxy under θXwhere θXis the
restriction of θto X, leaves of every star of H|Xunder θXare also consecutive under θ,
and for any given 1 ≤i < j ≤levery vertex of V(Si) is before every vertex of V(Sj) under
θ. We call this ordering a spiral galaxy ordering of Hand we call Haspiral galaxy under
θ. Let Q1, ..., Qtbe the stars of H|Xunder θX. If moreover all the following are satisfied:
•Centers of Siare incident to same number of leaves for i= 1, ..., l and |Si|=|Sj|for all
i6=j.
•t=land 2|Qi|=|Si| − 3 for i= 1, ..., l,
then we call Hauniform spiral galaxy under θ.
In this section we start by proving a structural property of α-coherent tournaments. Then
we prove that every spiral galaxy has the strong EH-property.
Lemma 3.1 Let c, α > 0be constants, where α≤c
m+1 . Let Tbe an α-coherent tournament
with |T|=n, and let A, B1, ..., Bmbe mvertex disjoint subsets of V(T)with |A| ≥ cn
and |Bi| ≥ cn for i= 1, ..., m. Then there exist vertices b1, ..., bm, a such that a∈A,
bi∈Bifor i= 1, ..., m, and {a}is complete to {b1, ..., bm}. Similarly there exist vertices
u1, ..., um, r such that r∈A,ui∈Bifor i= 1, ..., m, and {r}is complete from {u1, ..., um}.
Similarly there exist vertices s1, ..., st, ..., sm, r such that r∈A,si∈Bifor i= 1, ..., m, and
{s1, ..., st} ← {r} ← {st+1, ..., sm}.
Proof. We will prove only the first statement because the latter can be proved analogously.
Let Ai⊆Asuch that Aiis complete from Bifor i= 1, ..., m. Since Tis α-coherent, then
|Ai|< αn, and so A∗:= |A\(Sm
i=1 Ai)|≥ cn −mαn ≥(m+ 1)αn −mαn =αn. This implies
that A∗is non-empty. Fix a∈A. So there exist vertices b1, ..., bmsuch that {a}is complete
to {b1, ..., bm}.
The following well-known theorem will be very useful in our latter analysis:
Theorem 3.2 [8] Every tournament on 2k−1vertices contains a transitive subtournament
of size at least k.
5
An ordering αof the vertex set of a transitive tournament Sis called transitive ordering of
Sif AS(α) = φ. Let α′be an ordering of V(S). Define the transitive operation to be the
permutation of vertices that transforms α′to α(note possibly α=α′).
Let Hbe a regular path-galaxy tournament under an ordering θ= (u1, ..., uh) of its vertices.
Let Q1, ..., Qlbe the stars of Hunder θ. Let wbe an all-zero vector. We say that a smooth
(c, λ, w)-structure of a tournament Tcorresponds to Hif |w|=h. Define ΠH(θ) to be the
set of orderings of Hthat can be transformed to θby applying the transitive operation to
the leaves of all stars of H.
Let χ:= (A1, ..., A|w|) be a smooth (c, λ, w)-structure in a tournament Tthat corre-
sponds to H. We say that His well-contained in χif there exists xi∈Aifor i= 1, ..., |w|
such that T|{x1, ..., x|w|}is isomorphic to Hand (x1, ..., x|w|) is one of the orderings of H
in ΠH(θ).
Theorem 3.3 Every spiral galaxy has the strong EH-property.
Proof. Let Hbe a spiral galaxy tournament under an ordering θ= (u1, ..., uh) of its
vertices. We can assume that His a uniform spiral galaxy, since every spiral galaxy is a
subtournament of a uniform spiral galaxy. Let S1, ..., Slbe the pair-stars of Hunder θ, such
that |Si|= 2b+ 5. Let Tbe an H-free tournament on nvertices. Let ξ= 2b+ 5 and let
ν= 3l(2b−1+ 1). Let w:= (0, ..., 0) with |w|=ν(lξ + 1) + lξ be an all-zero vector. Theorem
2.2 implies that there exists c > 0 and a smooth (c, λ, w)-structure (A1, ..., A|w|) denoted
by χ, with λ=1
h. We are going to prove that there exists a pure pair (A, B) in Twith
O(A, B)≥αcn, where α=min{1
30(2b−1+2) ,b+1
h(2b−1+1) }. Assume that the contrary is true.
Then P(T)< αn. So Tis α-coherent. For k∈ {1, ..., l}, define Hk=H|Sk
j=1 V(Sj). Select
the sets Ai1, ..., Ailξ in χ, such that i1=ν+ 1, and ij+1 −ij=ν+ 1 for all j∈ {1, ..., lξ −1}.
Rename Ai1, ..., Ailξ by W1, ..., Wlξ. Clearly ˜χ= (W1, ..., Wlξ ) is a smooth (c, λ, ˜w)-structure,
where |˜w|=lξ. Let θk:= (uk1, ..., ukqk) be the restriction of θto V(Hk), where k∈ {1, ..., l}
and qk=k(2b+ 5) =|Hk|. For k=l, let q=ql. Rename now the vertices in the ordering
θkby v1, ..., vqk(that is θk:= (uk1, ..., ukqk) = (v1, ..., vqk)). Clearly Hkis a nebula under
θk. Let R1, ...., R2lbe the stars of Hlunder θl, and let L1, ..., L2lbe the set of leaves of
R1, ...., R2lrespectively. For all i∈ {1, ..., 2l}, let ai∈Libe a golden vertex of some pair
star. Define L′
i:= Li\{ai}and Eito be the set of arcs between L′
iand {ai}for i= 1, ..., 2l.
Let Hkbe the digraph obtained from Hkby deleting all the arcs in
2k
[
i=1
A(Hk|Li). When
k=l, write Hinstead of Hl. Let us call θkthe forest ordering of Hkand for i= 1, ..., 2l,
call L′
ia set of sister leaves of Hkunder θk(we call them sister leaves because they are
consecutive leaves under θ).
Claim 3.4 For some j1, ..., jqwith 1≤j1< ... < jq≤lξ, there exist vertices ai∈Wjifor
i= 1, ..., q, such that
•T′:= T|{a1, ..., aq}contains a copy of H, denoted by H′, and
•(a1, ..., aq)is the forest ordering of H, and
6
•every subtournament in T′induced by a set of sister leaves of H′under (a1, ..., aq), is
a transitive subtournament.
Proof of Claim 3.4. Assume that S1is a middle-pair-star and {v1, ..., vb} ← vb+1 (else
the argument is similar and we omit it). Let ρ= 2b−1. As α < c
ρ+2 and Wi≥cn
for i= 1, ..., |˜w|, Lemma 3.1 implies that there exist yi∈Wifor i= 1, ..., ρ + 1, ρ + 4,
such that {y1, ..., yρ} ← yρ+1 ←yρ+4. By Theorem 3.2, T|{y1, ..., yρ}contains a transitive
subtournament of size b. Let T1:= T|{yj1, ..., yjb}be this transitive subtournament, with
1≤j1< ... < jb≤ρ. Let W∗
ρ+2 ={y∈Wρ+2 :V(T1)∪ {yρ+1} → y→yρ+4}and
let W∗
i={y∈Wi:V(T1)∪ {yρ+1, yρ+4} → y}for i=ρ+ 5, ..., ξ. Lemma 2.3 implies
that |W∗
i| ≥ (1 −b+2
h)|Wi| ≥ |Wi|
2≥c
2nfor i=ρ+ 2, ρ + 5, ..., ξ. Now since α < c
2(ρ+2)
and W∗
i≥c
2nfor i=ρ+ 2, ρ + 5, ..., ξ, then by Lemma 3.1, there exist yi∈W∗
ifor
i=ρ+2, ρ+5, ..., ξ, such that yρ+2 ←yρ+5 ← {yρ+6, ..., yξ}. By Theorem 3.2, T|{yρ+6 , ..., yξ}
contains a transitive subtournament of size b. Let T2:= T|{yjb+6 , ..., yj2b+5 }be this transitive
subtournament. Let W∗
ρ+3 ={y∈Wρ+3 :V(T1)∪ {yρ+1, yρ+2} → y→ {yρ+4, yρ+5} ∪
V(T2)}. Similarly, we prove that W∗
ρ+3 ≥c
3n, and so W∗
ρ+3 6=φ. Fix yρ+3 ∈W∗
ρ+3. So
T|{yj1, ..., yjb, yρ+1, ..., yρ+5 , yjb+6, ..., yj2b+5 }contains a copy of Hl|V(S1). Denote this copy
by S1. Rename yj1, ..., yjb, yρ+1, ..., yρ+5, yjb+6, ..., yj2b+5 by a1, ..., a2b+5 respectively. Clearly
(a1, ..., a2b+5) is the forest ordering of Hl|V(S1), and T1and T2are transitive subtournaments
of T. If l= 1, we are done. So let us assume that l≥2.
Fix k∈ {1, ..., l −1}and let δ=k(2b+ 5). Assume that there exist vertices ai∈Wjifor
i= 1, ..., δ with 1 ≤j1< ... < jδ≤kξ, such that T|{a1, ..., aδ}contains a copy of Hkdenoted
by Hk′, (a1, ..., aδ) is the forest ordering of Hk, and every subtournament of Tinduced by a
set of sister leaves of Hk′under (a1, ..., aδ) is a transitive subtournament. Let t=k+ 1 and
for i=kξ + 1, ..., kξ +ξ, let W∗
i={y∈Wi:{a1, ..., aδ} → y}. By Lemma 2.3, |W∗
i| ≥ (1 −
δ
h)|Wi| ≥ (1−1
2)|Wi| ≥ |Wi|
2≥c
2nif l= 2, |W∗
i| ≥ |Wi|
3≥c
3nif l= 3, and |W∗
i| ≥ (1−δ
h)|Wi|
≥(1 −4
5)|Wi| ≥ |Wi|
5≥c
5notherwise. Assume that Stis a right-pair-star and assume that
{vδ+1, ..., vδ+b} ← vδ+2b+1 (else the argument is similar, and we omit it). Let Y1:= {kξ +
1, ..., kξ +ρ, kξ +2b+1, kξ+2b+4}. Since α < c
5(ρ+2) and for all i∈Y1,W∗
i≥c
5n, then Lemma
3.1 implies that for all i∈Y1, there exist yi∈W∗
i, such that {ykξ+1, ..., yk ξ+ρ} ← ykξ+2b+1 ←
ykξ+2b+4 . By Theorem 3.2, T|{ykξ+1, ..., ykξ+ρ}contains a transitive subtournament of size
b. Let T2k+1 := T|{yjδ+1 , ..., yjδ+b}be this transitive subtournament, with kξ + 1 ≤jδ+1 <
... < jδ+b≤kξ +ρ. Let Y2:= {kξ +ρ+ 1, ..., kξ + 2b, kξ + 2b+ 2, kξ +ξ}. For all
i∈Y2, define W∗∗
i:= \
x∈A
W∗
i,x, where A=V(T2k+1)∪ {ykξ+2b+1, ykξ +2b+4}. Then by
Lemma 2.3, for all i∈Y2,|W∗∗
i| ≥ (1 −5(b+2)
h)|W∗
i| ≥ |W∗
i|
6≥c
30 n. So as α < c
30(ρ+2) ,
then Lemma 3.1 implies that for all i∈Y2, there exist vertices yi∈W∗∗
i, such that
{ykξ+ρ+1, ..., ykξ+2b, ykξ +2b+2} ← ykξ+ξ. By Theorem 3.2, T|{ykξ +ρ+1, ..., ykξ+2b}contains a
transitive subtournament of size b. Let T2k+2 := T|{yjδ+b+1 , ..., yjδ+2b}be this transitive
subtournament, with kξ +ρ+ 1 ≤jδ+b+1 < ... < jδ+2b≤kξ + 2b. Now for i=kξ + 2b+ 3, let
W∗∗
i=\
x∈B
W∗
i,x, where B=V(T2k+1)∪V(T2k+2)∪ {ykξ+2b+1 , ykξ+2b+2 , ykξ+2b+4, ykξ+2b+5}.
Then by Lemma 2.3, |W∗∗
i| ≥ (1 −2
3)|W∗
i| ≥ |W∗
i|
3≥|Wi|
6if l= 2, |W∗∗
i| ≥ 2|W∗
i|
3≥
2|Wi|
9if l= 3, and |W∗∗
i| ≥ |W∗
i|
6≥|Wi|
30 if l≥4. Then W∗∗
i6=φ. Fix yi∈W∗∗
ifor
7
i=kξ + 2b+ 3. Rename yjδ+1 , ..., yjδ+2b, ykξ+2b+1 , ..., ykξ+2b+5 by aδ+1, ..., at(2b+5) respectively.
Now by merging G1:= T|{a1, ...., aδ}with G2:= T|{aδ+1, ..., at(2b+5)},G1∪G2contains a
copy of Htand (a1, ..., at(2b+5)) is its forest ordering. Also note that every subtournament
in G1∪G2induced by a set of sister leaves of Ht′under (a1, ..., at(2b+5)), is a transitive
subtournament.
Now applying this algorithm for t= 2, ..., l by turn, completes the proof. ♦
Let ϕ:= (a1, ..., al(2b+5)) and let γi:= (ai1, ..., ai2b+5 ) be the forest ordering of the copy Si
of H|V(Si) in T. Assume first that Siis a middle-pair-star. We know that {ai1, ..., aib}
is adjacent from pi
1∈ {aib+1 , aib+5 }. Let pi
2∈ {aib+1 , aib+5 }\{pi
1}. Note that pi
1and pi
2are
distinct. For j= 1,2, let qi
j∈ {aib+2 , aib+4 }, such that pi
jqi
j∈B(Si, γ). If {ai1, ..., aib} → qi
1
and qi
2→ {aib+6 , ..., ai2b+5 }, then do nothing. Otherwise: If pi
1=aib+1 , then there ex-
ist a1∈ {ai1, ..., aib, aib+2 },a2∈ {aib+1 , aib+5 },a3∈ {aib+4 , aib+6 , ..., ai2b+5 }, such that a1is
adjacent from {a2, a3}and a2is adjacent from a3. And if pi
1=aib+5 , then there exist
a1∈ {ai1, ..., aib+1 },a2∈ {aib+2 , aib+4 },a3∈ {aib+5 , ..., ai2b+5 }, such that a1is adjacent from
{a2, a3}and a2is adjacent from a3. In this case delete {ai1, ..., ai2b+5 }\{a1, a2, a3}from ϕ.
Assume now that Siis a right-pair-star. Let pi
1∈ {ai2b+1 , ai2b+5 }such that {ai1, ..., aib} ← pi
1
and let pi
2∈ {ai2b+1 , ai2b+5 }\{pi
1}. For j= 1,2, let qi
j∈ {ai2b+2 , ai2b+4 }, such that pi
jqi
j∈
B(Si, γi). If {ai1, ..., aib} → qi
1and {aib+1 , ..., ai2b} → qi
2, then do nothing. Else there exist
three vertices z1∈ {ai1, ..., ai2b},z2∈ {ai2b+1, ai2b+2 },z3∈ {ai2b+4 , ai2b+5 }, such that z1is ad-
jacent from {z2, z3}and z2is adjacent from z3. In this case delete {ai1, ..., ai2b+5 }\{z1, z2, z3}
from ϕ. Finally assume that Siis a left-pair-star. Let pi
1∈ {ai1, ai5}such that pi
1←
{ai6, aib+5 }and let pi
2∈ {ai1, ai5}\{pi
1}. For j= 1,2, let qi
j∈ {ai2, ai4}, such that
pi
jqi
j∈B(Si, γi). If qi
1→ {ai6, ..., aib+5 }and qi
2→ {aib+6 , ..., ai2b+5 }, then do nothing. Else
there exist r1∈ {ai1, ai2},r2∈ {ai4, ai5},r3∈ {ai6, ..., ai2b+5 }, such that r1is adjacent
from {r2, r3}and r2is adjacent from r3. In this case delete {ai1, ..., ai2b+5 }\{r1, r2, r3}from
ϕ. Now apply this algorithm for all i∈ {1, ..., l}. We get from ϕa new ordering, say
˜ϕ:= (ar1, ..., arf), with 3l≤f≤l(2b+ 5). Let ˆχ:= (Wr1, ..., Wrf). Clearly for i= 1, ..., f ,
ari∈Wri. Let ϕ∗be the ordering obtained from ˜ϕafter applying the transitive operation
to all sister leaves (if exist) of T|{ar1, ..., arf}under ˜ϕ.
Claim 3.5 There exists an ordering Σof H, satisfying all the following:
•V(H)is the disjoint union of V1and V2under Σ.
•H|V1is a path-galaxy under Σ1, the restriction of Σunder V1.
•T|{ar1, ..., arf}is a copy of H|V2,and ϕ∗is the ordering Σ2, where Σ2is the restriction of
Σunder V2.
Proof of Claim 3.5. Define operation Υ1to be the permutation of the vertices s1, ..., s5that
converts the ordering (s1, s2, s3, s4, s5) to the ordering (s4, s1, s3, s5, s2). Let ΘH(θ) be the set
of vertex orderings of Hthat are obtained from θby applying opperation Υ1to the golden
vertices of some pair-stars of Hunder θ. Clearly |ΘH(θ)|= 2l. Write ΘH(θ) := {θ1, ..., θ2l}.
Fix i∈ {1, ..., 2l}. For all ϑ∈ΘH(θ), let ϑPbe the restriction of Vto VP, where H|VP
is the path-galaxy under ϑPand with maximal order. Then there exist unique ordering
Σ∈ΘH(θ) such that V1=VPand T|{ar1, ..., arf}forms a copy of H|V2, where V2=V\VP.
And moreover ϕ∗is the ordering Σ2, the restriction of Σ to V2. This terminates the proof
8
of Claim 3.5. ♦
Let Σibe the restriction of Σ to Vifor i= 1,2. Write Σ = (m1, ..., mh), Σ1= (mg1, ..., mgκ),
Σ2= (mq1, ..., mqf). Let P:= H|V1. We know that Pis a uniform path-galaxy under Σ1.
Let Q1, ..., Qηbe the stars of Punder Σ1. Let H+be the tournament obtained from H
by replacing Qiby a star with 2b−1leaves for i= 1, ..., η. More formal speaking: For all
1≤i≤η, let mgi0be the center of Qiand mgi1, ..., mgibbe its leaves (clearly mgi1, ..., mgib
are consecutive under Σ). Let H+be the tournament with V(H+) = V(H)∪(
η
[
i=1
{ci
1, ..., ci
τ}
such that E(B(H+,Σ+)) = E(B(H, Σ))∪(
η
[
i=1
{mgi0ci
1, ..., mgi0ci
τ}), where Σ+is the ordering
obtained from Σ after inserting ci
1, ..., ci
τjust before mgi1in Σ for all 1 ≤i≤η, where
τ= 2b−1−b. Rename Σ+as follows: Σ+= (n1, ..., nh+ητ ). Let Σ+
2:= (nh1, ..., nhf) be the
restriction of Σ+to V2. Rename Wriby Nhifor i= 1, ..., f. Now enrich ˆχ= (Nh1, ..., Nhf)
by p:= κ+ητ sets from χafter renaming them by Ne1, ..., Nep, in a way that the outcome
will be (N1, ..., Nh+ητ ) and ˆχ+:= (N1, ..., Nh+ητ ) is a smooth (c, 1
h, w)-structure, where w
is an all-zero vector of length h+ητ. We can do that because for any i∈ {1, ..., f −1},
there exist in χat least νsets lying between Nhiand Nhi+1. Clearly Σ+
1= (ne1, ..., nep) is
the restriction of Σ+to V+
1:= V(H+)\V2, and H+
1:= H+|V+
1is a path-galaxy under Σ+
1.
Let Q+
1, ..., Q+
ηbe the stars of H+
1under Σ+
1. For simplicity, for i= 1, ..., p, let us rename
Neiand neiby Eiand wirespectively. For all 1 ≤i≤η, let wiobe the center of Q+
iand
wi1, ..., wiρbe its leaves.
Claim 3.6 For all 1≤i≤η, there exist bsets Eiι1, ..., Eiιb, with i1≤iι1< ... < iιb≤iρ,
such that Pis well-contained in χ∗, the smooth (c, λ, w∗)-structure containing the sets in
E:=
i=η
[
i=1
{Ei0, Eiι1, ..., Eiιb}(note that the sets of Eare ordered in χ∗according to their
appearance in χ), where w∗is an all-zero vector of length η(b+ 1).
Proof of Claim 3.6. Let Y:= {ar1, ..., arf}and for i= 10, ..., 1ρ, let E∗
i=\
x∈Y
Ei,x. By
Lemma 2.3, for all i∈ {10, ..., 1ρ},|E∗
i| ≥ (1 −f
h)|Ei| ≥ 2|Ei|
9≥2c
9n. Denote 2c
9by
ˆc. For simplicity, for i= 1, ..., κ, let us rename mgiby di. For k∈ {1, ..., η}, define
Pk=P|Sk
j=1 V(Qj), where Pη=P. The proof works as follows: we will construct P
star by star after updating the sets corresponding to each star in order to merge it with the
previously constructed stars. Let Σk
1:= (dk1, ..., dkq) be the restriction of Σ1to V(Pk), where
k∈ {1, ..., η}and q=k(b+ 1) =|Pk|. As α < ˆc
ρ+1 and E∗
1i≥ˆcn for i= 0, ..., ρ, then Lemma
3.1 implies that there exist x1i∈E∗
1ifor r= 0, ..., ρ, such that x10← {x11, ..., x1ρ}if Q+
1is
a left star, and {x11, ..., x1ρ} ← x10if Q+
1is a right star. By Theorem 3.2, T|{x11, ..., x1ρ}
contains a transitive subtournament of size b. So there exists bvertices x1ιi∈E1ιifor
i= 1, ..., b, with 11≤1ι1< ... < 1ιb≤1ρ, and such that T|{x1ι1, ..., x1ιb}is a transitive
subtournament. Let Λ1be the ordering of {x10, x1ι1, ..., x1ιb}according to their appearance
in χ. Then by applying the transitive operation to {x1ι1, ..., x1ιb}transforms Λ1to the
9
star ordering of Q1(i.e the restriction Σ1
1of Σ1to V(Q1)). Hence Λ1∈ΠP1(Σ1
1). Fix
k∈ {1, ..., η −1}and assume that for i= 1, ..., k, there exist bsets Eiι1, ..., Eiιb, with
i1≤iι1< ... < iιb≤iρ, such that Pkis well-contained in χ∗
k, the smooth (c, λ, w∗
k)-structure
containing the sets in E:=
i=k
[
i=1
{Ei0, Eiι1, ..., Eiιb}. Denote by Pkthe well-contained copy
of Pk. Let t=k+ 1. For i= 0, ..., ρ, let E∗
ti=\
x∈Y∪Pk
Eti,x. By Lemma 2.3, for all
i∈ {0, ..., ρ},|E∗
ti| ≥ (1−f+k(b+1)
h)|Eti| ≥ (b+1)|Eti|
h≥(b+1)c
hn. As α < (b+1)c
h(ρ+1) and E∗
ti≥b+1
hcn
for i= 0, ..., ρ, then Lemma 3.1 implies that there exist xti∈E∗
tifor i= 0, ..., ρ, such
that xt0← {xt1, ..., xtρ}if Q+
tis a left star, and {xt1, ..., xtρ} ← xt0if Q+
tis a right star.
By Theorem 3.2, T|{xt1, ..., xtρ}contains a transitive subtournament of size b. So there
exists bvertices xtιi∈Etιifor i= 1, ..., b, with t1≤tι1< ... < tιb≤tρ, and such that
T|{xtι1, ..., xtιb}is a transitive subtournament. Now merging Pkwith T|{xt0, xtι1, ..., xtιb}
we get a well-contained copy of Pkand moreover the ordering of the vertices of Ptaccording
to their appearance in χbelongs to ΠPt(Σt
1), where Ptis the well-contained copy of Pt, and
Σt
1is the restriction of Σ1to V(Pt). Applying this algorithm for t= 2, ..., η completes the
proof. ♦
Now by merging the well-contained copy of Pwith T|Ywe get a tournament in Twhich is
isomorphic to H. So Tcontains H, a contradiction. This completes the proof.
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