Almost partitioning 2-coloured complete 3-uniform hypergraphs into two monochromatic tight or loose cycles

0.17 MB·

by  Sebastian Bustamante

Almost partitioning 2-coloured complete 3-uniform
7
1 hypergraphs into two monochromatic tight or loose
0
2
cycles

n
a
J
Sebasti´an Bustamante∗†,1, Hiˆe.p H`an∗,‡,2, and Maya Stein∗,§,1
6
2
1Department of Mathematical Engineering, University of Chile

] 2Instituto de Matem´aticas, Pontificia Universidad Cat´olica de
O
Valpara´ıso
C
.
h
t 27th January 2017
a
m
[

1
v Abstract
6
0 We show that for every η > 0 there exists an integer n0 such that
8 every 2-colouring of the 3-uniform complete hypergraph on n n0
≥
7 vertices contains two disjoint monochromatic tight cycles of distinct
0
colours that together cover all but at most ηn vertices. The same
.
1 result holds if we replace tight cycles with loose cycles.
0
7
1 1 Introduction
:
v
i
X A certain type of Ramsey problems concerned with covering all vertices of
the host graph instead of finding a small subgraph has gained popularity in
r
a
recent years, bothforgraphsandhypergraphs. Inparticular, theproblemof
partitioning an edge-coloured complete (hyper-)graph into monochromatic
cycles has received much attention. The recent surveys [5, 6] are a good
starting point for the vast literature on the subject.
∗All three authors acknowledge support by Millenium Nucleus Information and Co-
ordination in Networks ICM/FIC RC130003.
†Thefirst author was supported by CONICYT Doctoral Fellowship grant 21141116.
‡Thesecond author was supported byFondecyt Iniciaci´on grant 11150913.
§Thethird authorreceived support by Fondecyt Regular grant 1140766.
1
Central to the area is an old conjecture of Lehel for graphs (see [2]). It
states that for every n, every two-colouring of the edges of the complete
graph K admits a covering of the vertices of K with at most two mono-
n n
chromatic vertex disjoint cycles of different colours. (For technical reasons,
a single vertex or an edge count as a cycle.) This was confirmed for large n
in[13,1],andforallnbyBessyandThomass´e[3]. Thereisplentyofactivity
in determing the number of monochromatic cycles needed if K is coloured
n
with more than two colours. It is known that this number is independent
of n, but the best known lower and upper bounds, r + 1 and 100rlogr,
respectively, leave a considerable gap [15, 7].
For hypergraphs, much less is known. The problems transforms in the
obvious way to hypergraphs, considering r-edge-colourings of the k-uniform
(k)
on n vertices, the only question is which type of hypergraph cycles we
n
K
would like to work with. Referring to [5] for other results, we concentrate
here on loose and tight cycles (see the next section for their definitions).
For loose cycles the problem was studied in [8, 17] and the best bound,
due to Sa´rk¨ozy [17], shows that at most 50rklog(rk) disjoint loose mono-
(k)
chromatic cycles are needed for partitioning , a number that this is
n
K
independent of n. In general even for the case k = 3 the problem is wide
openanditisforexampleunknownwhetheronecancoveralmostallvertices
with two disjoint monochromatic loose cycles.
Concerning the more restrictive notion of tight cycles, the situation is
even worseandtoourbestknowledge, nothingis known. Gy´arf´as [6]conjec-
(k)
tures that there is c =c(r,k) such that every r-coloured has a partition
n
K
into at most c monochromatic tight cycles, but this is open even for the
‘easiest’ case of 3-uniform hypergraphs and two colours. Our main result
establishes an approximate version for this case.
Theorem 1.1. For every η > 0 there exists n such that if n n then
0 0
≥
(3)
every two-coloring of the edges of the complete 3-uniform hypergraph
n
K
admits two vertex-disjoint monochromatic tight cycles, of distinct colours,
which cover all but at most ηn vertices.
Moreover, we can choose the parity of the length of each of the cycles.
Wemightbeinterested inchoosingtheparityofthecyclesforthefollow-
ing reason. If ℓ is even, then any 3-uniform tight cycle on ℓ edges contains
a loose cycle. Hence, we can deduce that an analogue of Theorem 1.1 holds
for loose cycles.
Corollary 1.2. For every η > 0 there exists n such that if n n then
0 0
≥
(3)
every two-coloring of the edges of the complete 3-uniform hypergraph
n
K
2
admits two vertex-disjoint monochromatic loose cycles, of distinct colours,
which cover all but at most ηn vertices.
We believe that the error term ηn in Theorem 1.1 can be improved and
(3)
that every two-colouring of the edges of admits two disjoint monochro-
n
K
matictightcycleswhichcover allbutatmostaconstantnumbercofvertices
(for some c independent of n).
The proof of Theorem 1.1 is inspired by the work of Haxell et al. [10, 11]
and relies on an application of the hypergraph regularity lemma [4]. This
reduces the problem at hand to that of finding, in any two-colouring of the
edges of an almost complete 3-uniform hypergraph, two disjoint monochro-
matic connected matchings which cover almost all vertices.
Here, as usual, a matching in hypergraph is a set of pairwise
M H
disjointedgesand iscalled connected ifbetweenevery paire,f
M⊂ H ∈ M
there is a pseudo-path in connecting e and f, that is, there is a sequence
H
(e ,...,e ) of not necessarily distinct edges of such that e = e ,f = e
1 p 1 p
H
and e e = 2 for each i [p 1]. (Note that these pseudo-paths may
i i+1
| ∩ | ∈ −
use vertices outside V( ).) Now, we call a connected matching in a 2-
M M
coloured hypergraph a monochromatic connected matching if all edges in
M
and all edges on the connecting paths have the same colour.
So, our main contribution reduces to the following result, which might
be of independent interest.
Theorem 1.3. For every γ > 0 there is t such that the following holds
0
for every 3-uniform hypergraph with t > t vertices and (1 γ) t edges.
H 0 − 3
Any two-colouring of the edges of admits two disjoint monoc(cid:0)h(cid:1)romatic
H 1
connected matchings covering at least (1 240γ6)t vertices of .
− H
We prove Theorem 1.3 in Section 2. In Section 3, we introduce the
regularity lemma for hypergraphs and state an embedding result from [11].
The proof of Theorem 1.1 will be given in Section 4.
2 Monochromatic connected matchings
Before giving the proof of Theorem 1.3 we introduce some notation and
auxiliary results.
Let denote a k-uniform hypergraph, that is, a pair = (V,E) with
H H
finite vertex set V = V( ) and edge set E = E( ) V , where V
H H ⊂ k k
denotes the set of all k-element sets of V. Often will be(cid:0)id(cid:1)entified w(cid:0)ith(cid:1)
H
its edges, that is, V and for an edge x ,...,x we often omit
H ⊂ k { 1 k} ∈ H
brackets and write x .(cid:0)..(cid:1)x only. A k-uniform hypergraph is called an
1 k
C
3
ℓ-cycle if there is a cyclic ordering of the vertices of such that every edge
C
consists of k consecutive vertices, every vertex is contained in an edge and
two consecutive edges (where the ordering of the edges is inherited by the
ordering of the vertices) intersect in exactly ℓ vertices. For ℓ = 1 we call
the cycle loose whereas the cycle is called tight if ℓ = k 1 (and we do not
−
consider other values of ℓ).
A tight path is a cycle from which one vertex and all incident edges are
deleted. Thelength ofapath,apseudo-pathoracycleisthenumberofedges
itcontains. Asabove, two edgesin areconnectedifthereisapseudo-path
H
connecting them. Connectedness is an equivalence relation on the edge set
of and the equivalence classes are called connected components.
H
All hypergraphs considered from now on are 3-uniform. We will
H
need the following result concerning the existence of perfect matchings in
3-uniform hypergraphs with high minimum vertex degree.
Theorem 2.1 ([9]). For all η > 0 there is a n = n (η) such that for all
0 0
n > n , n 3Z, the following holds. Suppose is 3-uniform hypergraph on
0
∈ H
n vertices such that every vertex is contained in at least 5 +η n edges.
9 2
Then contains a perfect matching. (cid:0) (cid:1)(cid:0) (cid:1)
H
Denote by ∂ the shadow of , that is, the set of all pairs xy for which
H H
there exists z such that xyz . For a vertex x in a hypergraph , let
∈ H H
N (x) = y : xy ∂ . Fortwoverticesx,y,letN (x,y) = z : xyz .
H H
{ ∈ H} { ∈ H}
Note that if y N (x) (equivalently, x N (y)) then N (x,y) = . We
H H H
∈ ∈ 6 ∅
call all such pairs xy of vertices active.
Lemma 2.2 ([10], Lemma 4.1). Let γ > 0 and let be a 3-uniform hy-
H
pergraph on t vertices and at least (1 γ) t edges. Then contains
H − 3 H
a subhypergraph on t (1 10γ1/6)t (cid:0)ve(cid:1)rtices such that every ver-
K H
K ≥ −
tex x of is in an active pair of and for all active pairs xy we have
K K
N (x,y) (1 10γ1/6)t .
K K
| | ≥ −
We are now ready to prove Theorem 1.3.
Proof of Theorem 1.3. For given γ > 0 let δ = 10γ1/6 and apply The-
orem 2.1 with η = 1/9 to obtain n . We choose t = max 2, n0 .
0 0 δ 27δ
Suppose we are given a two-coloured 3-uniform hyperg(cid:8)raph (cid:9)=
red
H H ∪
ont > t vertices and(1 γ) tH edges. ApplyLemma2.2to with
Hblue H 0 − 3 H
parameter γ to obtain , t := t w(cid:0)ith(cid:1)the properties stated in the lemma.
K
K
We wish to find two monochromatic connected matchings covering all but
at most 24δt 24δt vertices of .
H
≤ K
4
Since every vertex is in an active pair in , we have
K
N (x) (1 δ)t for all x V( ). (1)
K
| | ≥ − ∈ K
Let = be the colouring of inherited from . Then a
red blue
K K ∪ K K H
monochromatic component of is a connected component or .
red blue
C K K K
Observation 2.3 ([11], Observation 8.2). For every vertex x V( ) there
∈ K
exists a monochromatic component such that N (x) (1 δ)t.
Cx | Cx | ≥ −
For each x V( ) choose arbitrarily one component as in Observa-
x
∈ K C
tion 2.3. Let R = x V( ) : is red and B = x V( ) : is blue ,
x x
{ ∈ K C } { ∈ K C }
and note that these two sets partition V( ).
K
Observation 2.4 ([11], Observation 8.4). If R 6δt (or B 6δt, re-
| | ≥ | | ≥
spectively), then there is a red component (a blue component ) such that
R B
= ( = ) for all but at most 2δt vertices x R (x B).
x x
C R C B ∈ ∈
Set V := x R : C = if V 6δt, and set V := x
red x red blue
{ ∈ R} | | ≥ { ∈
B : C = if B 6δt. Otherwise let V , or V , respectively, be the
x red blue
B} | | ≥
empty set. Our aim is to find two differently coloured disjoint connected
matchings in that together cover all but 12δt 24δt 12δt vertices of
K ≤ −
V V .
red blue
∪
We start by choosing a connected matching of maximal size in
R ∪
. This matching decomposes into two disjoint monochromatic connected
B
matchings, and ,whichtogethercoverasmanyvertices
red blue
M ⊂ R M ⊂ B
as possible. Let V′ = V V( ) and V′ = V V(
red red\ Mred∪Mblue blue blue\ Mred∪
). We may assume that V′ or V′ has at least 12δt vertices, as
Mblue red blue
otherwise we are done. By symmetry we may assume that
V′ 12δt. (2)
| red| ≥
Observe that there is no edge xy with x V′ and y V′ such that
∈ red ∈ blue
xy ∂ ∂ . Indeed, any such edge xy constitutes an active pair (by
∈ R ∩ B
Lemma 2.2) and as V′ > δt +2, there must be a vertex z V′ such
| red| ∈ red
that xyz is an edge of . This contradicts the maximality of the matching
K
.
red blue
M ∪M
Next, we claim that
V′ 2δt. (3)
| blue|≤
Assume otherwise. Then, Observation 2.3 and the choice of the set V
red
implies that the number of edges between V′ and V′ that belong to ∂
red blue R
is at least
1
V′ (V′ δt) V′ V′ .
| red|· | blue|− ≥ 2| red|·| blue|
5
Similarly,thereareatleast V′ (V′ δt) > 1 V′ V′ edgesbetween
| blue|· | red|− 2| red|·| blue|
V′ and V′ that belong to ∂ . As there is no edge xy with x V′ and
red blue B ∈ red
y V′ such that xy ∂ ∂ , we have more than V′ V′ edges
∈ blue ∈ R∩ B | red|·| blue|
from V′ to V′ . This yields a contradiction and (3) follows.
red blue
Because of the maximality of , each edge having all its
red blue
M ∪ M
vertices in V′ is blue. Fix one such edge xyz. Obtain V′′ from V′ by
red red red
deleting the at most δt vertices w with wx / ∂ . Consider any edge x′y′z′
∈ R
withx′,y′,z′ V′′ . Asthepairsxy,xx′,x′y′ areallactive, and V′′ > 3δt,
∈ red | red|
there is a vertex v V′′ that forms an edge with each of the three pairs,
∈ red
thus giving a pseudo-path in [V′′ ] from xyz to x′y′z′. Denote by ′′ the
K red B
blue component of [V′′ ] that contains xyz, and let ′ be obtained from
K red B
′′ by deleting at most 2 vertices and all incident edges, so that V[ ′] is a
B | B |
multiple of 3. Then, by (2), we have
V[ ′] V′ δt 2 10δt. (4)
| B |≥ | red|− − ≥
Let x V[ ′] be given. At least V[ ′] δt vertices y V[ ′] are such
∈ B | B |− ∈ B
that xy ∂ , and, for each such y there are at least V[ ′] δt vertices
∈ R | B |−
z V[ ′] such that xyz ′. So, the total number of hyperedges of ′ that
∈ B ∈ B B
contain x is at least
1 1 9 2 2 V[ ′]
(V[ ′] δt)2 V[ ′] | B | .
2 | B |− ≥ 2(cid:18)10| B |(cid:19) ≥ 3(cid:18) 2 (cid:19)
Thus, Theorem 2.1 with η = 1 yields a perfect matching ′ of ′.
9 Mblue B
At this point, we have three disjoint monochromatic connected match-
ings, one in red ( ) and two in blue ( and ′ ′).
Mred ⊂ R Mblue ⊂ B Mblue ⊂ B
Together, thesematchingscover allbutatmost3δt+2 vertices ofV V
red blue
∪
(by (3) and by (4)). In particular, we can assume that contains at
blue
M
least two hyperedges, as otherwise we can just forget about and are
blue
M
done.
Ouraimisnowtodissolvethebluematching ,andcoveritsvertices
blue
M
by new red edges, leaving at most 6δt vertices uncovered. In order to do so,
let us first understand where the edges of lie.
blue
M
For convenience, let us call an edge in good if two different pairs of its
K
vertices a,b and c,d are such that ab ∂ and cd ∂ . Notice that
{ } { } ∈ R ∈ B
every good red edge is contained in and every good blueedge is contained
R
in .
B
First, we claim that for every edge uvw ,
blue
∈ M
u,v,w V 1. (5)
blue
|{ }∩ |≤
6
Indeed, otherwise there is an edge uvw with u,v V . By
blue blue
∈ M ∈
thedefinitionof ,andby (2),thereisanactive edgeua ∂ witha V′ .
B ∈ B ∈ red
As uais an active pair, as a has very large degree in ∂ , andby (2), there is
R
an edge uab with b V′ such thatab ∂ . Hence uab is agood edge.
∈ K ∈ red ∈ R
Similarly, there is a good edge vcd, with c,d V′ a,b . Remove the
∈ red \ { }
edge uvw from and add edges uab and vcd to either or ,
blue red blue
M M M
according to their colour. The resulting matching covers more vertices than
the matching M , a contradiction. This proves (5).
A blue
∪M
Next, we claim that there is no edge uvw with
blue
∈M
u,v,w V = 1. (6)
blue
|{ }∩ |
Assume otherwise. Then there is an edge uvw with u V
blue blue
∈ M ∈
and v,w V . As in the proof of (5), we can cover u with a good edge uab
red
∈
such that a,b V′ . Moreover, since vw is an active pair, and v has very
∈ red
large degree in ∂ , there is an edge vwc with c V′ a,b and cv ∂R.
R ∈ red\{ } ∈
Since vw ∂ , the edge vwc is good. So we can remove uvw from
blue
∈ B M
and add edges uab and vwc to , thus covering three additional
red blue
M ∪M
vertices. This gives the desired contradiction to the choice of ,
red blue
M ∪M
and proves (6).
Putting (5) and (6) together, we know that for every edge uvw
blue
∈M
we have u,v,w V . Consider any two edges u v w ,u v w . As
red 1 1 1 2 2 2 blue
∈ ∈ M
before, there are vertices a,b V′ such that edges v w a,v w b are good.
∈ red 1 1 2 2
Now, if there is a red edge u u c with c V′ and u c ∂ then we can
1 2 ∈ red 1 ∈ R
remove edges u v w ,u v w andadd thered edgeu u c to andedges
1 1 1 2 2 2 1 2 red
M
v w a,v w b to , according to their colour, contradicting the
1 1 2 2 red blue
M ∪M
choiceof . Therefore,foranychoiceofu v w ,u v w ,
red blue 1 1 1 2 2 2 blue
M ∪M ∈ M
we have that
all edges u u c with c V′ and u c ∂ are blue. (7)
1 2 ∈ red 1 ∈ R
Moreover, if there is a blue edge u u x with x v ,w ,v ,w then
1 2 1 1 2 2
∈ { }
u u is an active pair. In that case, we can calculate as before that an edge
1 2
u u c with c V′ and u c ∂ exists, and by (7), this edge is blue. The
1 2 ∈ red 1 ∈ R
existence of the blue edge u u x implies that we can link u u c to
1 2 1 2 blue
M
with a blue tight path. Thus, removing u v w and u v w from and
1 1 1 2 2 2 blue
M
adding v w a,v w b,u u c to (where a,b are as above), we
1 1 2 2 1 2 red blue
M ∪M
obtain a contradiction to the choice of . So, for any choice of
red blue
M ∪M
u v w ,u v w , we have that
1 1 1 2 2 2 blue
∈ M
all edges u u x with x v ,w ,v ,w are red. (8)
1 2 1 1 2 2
∈ { }
7
We can now dissolve the edges of . For this, separate each hy-
blue
M
peredge uvw in into an edge uv and a single vertex w. Let X be the
blue
M
set of all edges uv, and let Y be the set of all vertices w obtained in this
way. Note that every uv X is an active pair in , and therefore forms a
∈ K
hyperedge uvw′ with all butat most δt of the vertices w′ Y. Moreover, all
∈
but at most δt of these hyperedges uvw′ are such that uw′ ∂ , because
∈ R
of the large degree u has in ∂ .
R
ConsiderthebipartitegraphonX Y whichhasanedgebetweenuv X
∪ ∈
and w′ whenever the hyperedge uvw′ exists in and uw′ ∂ . Then for
K ∈ R
each X′ X we have that N(X′) Y 2δt X′ 2δt. So by the
⊆ | | ≥ | |− ≥ | |−
defect form of Hall’s Theorem, there is a matching covering all but at most
2δt vertices of X′ . In , this corresponds to a matching ′ covering all
| | K Mred
but at most 6δt vertices of V( ). By (8), all hyperedges of ′ are
Mblue Mred
red. Furthermore, since we ensured that every hyperedge in contains
red
M
a pair uw′ that forms an ede of ∂ , we know that and ′ belong
R Mred Mred
to the same red component of . In other words, ′ and ′
K Mred ∪Mred Mblue
are the two monochromatic connected matchings we had to find.
3 Hypergraph regularity
In this section we introducetheregularity lemma for 3-uniform hypergraphs
and state an embedding result from [11].
Graph regularity. Let G be a graph G and let X,Y V(G) be disjoint.
⊆
The density of (X,Y) is d (X,Y) = eG(X,Y) where e (X,Y) denotes the
G |X||Y| G
number of edges of G between X and Y.
The bipartite graph G on the partition classes X and Y is called (d,ε)-
regular, if d (X′,Y′) d < ε holds for all X′ X and Y′ Y of size
G
| − | ⊆ ⊆
X′ > εX and Y′ > εY . If d= d (X,Y) we say that G is ε-regular.
G
| | | | | | | |
Hypergraph regularity. Let be a 3-uniform hypergraph. Let P =
H
P12 P13 P23 with V(P) V( ) be a tripartite graph which we also
∪ ∪ ⊂ H
refer to as triad. By (P) denote the 3-uniform hypergraph on V(P) whose
T
edges are the triangles of P. The density of with respect to P is
H
(P)
d (P) = |H∩T |.
H
(P)
|T |
8
Similarly, for a tuple ~ = (Q ,...,Q ) of subgraphs of P, we define the
1 r
Q
density of with respect to ~ as
H Q
(Q )
d (~)= |H∩ i∈[r]T i |.
H Q S (Q )
| i∈[r]T i |
S
Letα,δ > 0andletr > 0beaninteger. Wesaythat is(α,δ,r)-regular
H
with respect to P if, for every r-tuple ~ = (Q ,...,Q ) of subgraphs of P
1 r
Q
satisfying (Q ) > δ (P), we have d (~) α < δ. If α = d (P)
| i∈[r]T i | |T | | H Q − | H
we say thatS is (δ,r)-regular with respect to P, and in the same situation,
H
we say P is (δ,r)-regular (with respect to ).
H
Moreover, ifthebipartitegraphsP12,P13,P23 ofan(α,δ,r)-regular P =
P12 P13 P23 are (1/ℓ,ε)-regular then we say that the pair ( ,P) is an
∪ ∪ H
(α,δ,ℓ,r,ε)-regular complex.
Finally, a partition of V into V V V is called an equipartition
0 1 t
∪ ∪···∪
if V < t and V = V = = V .
0 1 2 t
| | | | | ··· | |
Westatetheregularitylemmafor3-uniformhypergraphs[4]aspresented
in [14].
Theorem 3.1 (Regularity Lemma for 3-uniform Hypergraphs). For all
δ,t > 0, all integer-valuedfunctions r = r(t,ℓ), and alldecreasing sequences
0
ε(ℓ) > 0 there exist constants T ,L and N such that every 3-uniform hy-
0 0 0
pergraph with at least N vertices admits a vertex equipartition
0
H
V( )= V V V with t t < T ,
0 1 t 0 0
H ∪ ∪···∪ ≤
and, for each pair i,j, 1 i < j t, an edge partition of the complete
≤ ≤
bipartite graph
ij
K(V ,V ) = P with 1 ℓ < L
i j k ≤ 0
[
k∈[ℓ]
such that
ij
1. all graphs P are (1/ℓ,ε(ℓ))-regular.
k
2. is (δ,r)-regular with respect to all but at most δℓ3t3 tripartite graphs
H
Phi Phj Pij.
a ∪ b ∪ c
Note that the same partitions satisfy the conclusions of Theorem 3.1
for the complement of as well. Further, as noted in [11] by choosing a
H
random index k [ℓ] for each pair (V ,V ) Markov’s inequality yields that
ij i j
∈
with positive probability there are less than 2δt3 chosen triads which fail to
be (δ,r)-regular. Hence one obtains the following.
9
Observation3.2. Inthepartition produced byTheorem 3.1there isafamily
of bipartite graphs Pij = Pij with vertex classes V ,V , where 1 i < j
P kij i j ≤ ≤
t, such that is (δ,r)-regular with respect to all but at most 2δt3 tripartite
H
graphs Phi Phj Pij.
∪ ∪
We end this section with a result from [16] and [11] which allows embed-
ding tight paths in regular complexes. In the following, an S-avoiding tight
path is one which does not contain any vertex from S. (Note that although
Lemma 4.6 from [11] is stated slightly differently, its proof actually yields
the version below.)
Lemma 3.3 ([11], Lemma 4.6). For each α (0,1) there exist δ > 0 and
1
∈
sequences r(ℓ), ε(ℓ), and n (ℓ), for ℓ N, with the following property.
1
∈
For each ℓ N, and each δ δ , if ( ,P) is a (d (P),δ,ℓ,r(ℓ),ε(ℓ))-
1 H
∈ ≤ H
complex with d (P) α and all of the three vertex classes of P have the
H
≥
same size n > n (ℓ), then there is a subgraph P on at most 27√δn2/ℓ edges
1 0
ofP suchthat, forallordered pairs ofdisjointedges(e,f) (P P ) (P P )
0 0
∈ \ × \
there is m = m(e,f) [3] such that the following holds. For every S
∈ 1 ⊆
V( ) (e f) with S < n/(logn)2, and for each ℓ with 3 ℓ (1 2δ4)n,
H \ ∪ | | ≤ ≤ −
there is an S-avoiding tight path from e to f of length 3ℓ+m in .
H
4 Proof of Theorem 1.1
We follow a procedure suggested by L uczak in [12] for graphs and used for
tight cycles in 3-uniform hypergraphs in [11].
Proof of Theorem 1.1. For given η > 0 we apply Lemma 1.3 with γ =
(η/480)6 to obtain t . With foresight apply Lemma 3.3 with α = 1/2 to
0
obtain δ , and sequences r(ℓ), ε(ℓ), and n (ℓ). Finally, apply Theorem 3.1
1 1
witht ,r(t,ℓ) = r(ℓ),ε(ℓ), n (ℓ)andδ = min δ /2,γ/48,(η/16)4 toobtain
0 1 1
{ }
constants T , L and N .
0 0 0
Given a two-colouring = of the 3-uniform complete
n red blue
K H ∪ H
hypergraph on n > N vertices. Apply Theorem 3.1 with the chosen
n 0
K
constants to to obtain partitions
red
H
ij
V( )= V V V and K(V ,V ) = P ,1 i < j t
Kn 0∪ 1∪···∪ t i j k ≤ ≤
[
k∈[ℓ]
with t t < T , and ℓ < L which satisfy the properties detailed in
0 0 0
≤
Theorem 3.1. The partitions satisfy the same properties for as it is
blue
H
the complement hypergraph of .
red
H
10