Turan Problems on Non-uniform Hypergraphs

0.34 MB·English

by  Travis Johnston

Tur´an Problems on Non-uniform Hypergraphs
3
Travis Johnston ∗ Linyuan Lu †
1
0
2 January 10, 2013

n
a
J
9 Abstract
A non-uniform hypergraph H = (V,E) consists of a vertex set V and an edge set E ⊆ 2V;
]
O the edges in E are not required to all have the same cardinality. The set of all cardinalities of
edges in H is denoted by R(H), the set of edge types. For a fixed hypergraph H, the Tur´an
C
. density π(H) is defined to be limn→∞maxGnhn(Gn), where the maximum is taken over all H-
h free hypergraphs Gn on n vertices satisfying R(Gn) ⊆ R(H), and hn(Gn), the so called Lubell
at function, istheexpectednumberof edgesin Gn hitbyarandom fullchain. This concept,which
m generalizestheTur´andensityofk-uniformhypergraphs,ismotivatedbyrecentworkonextremal
poset problems. Thedetails connectingthesetwo areas will berevealed intheendof thispaper.
[
Several properties of Tur´an density, such as supersaturation, blow-up, and suspension, are

1 generalized from uniform hypergraphs to non-uniform hypergraphs. Other questions such as
v “Whichhypergraphsaredegenerate?” aremorecomplicated anddon’tappeartogeneralize well.
0 In addition, we completely determinethe Tur´an densities of {1,2}-hypergraphs.
7
8
1 1 Introduction
.
1
0 AhypergraphH isapair(V,E); V isthe vertexset,andE 2V isthe edgeset. Ifalledgeshavethe
⊆
3 same cardinality k, then H is a k-uniform hypergraph. Tur´an problems on k-uniform hypergraphs
1
have been actively studied for many decades. However,Tur´anproblems on non-uniform hypergraphs
:
v are rarely considered (see [33, 30] for two different treatments). Very recently, several groups of
i peoplehavestartedactivelystudyingextremalfamiliesofsetsavoidinggivensub-posets. Severalnew
X
problems have been established. One of them is the diamond problem:
r
a The diamond conjecture: [23]Anyfamily ofsubsetsof[n]:= 1,2,...,n withnofoursets
A,B,C,D satisfying A B C, B C D canFhave at most (2+o(1{)) n sub}sets.
⊆ ∩ ∪ ⊆ ⌊n2⌋
This conjecture, along with many other problems, motivates us to study Tur´an-type problems on
(cid:0) (cid:1)
non-uniform hypergraphs. The details of this connection will be given in the last section.
WebrieflyreviewthehistoryofTur´anProblemsonuniformhypergraphs. Givenapositiveinteger
n and a k-uniform hypergraph H on n vertices (or k-graph, for short), the Tur´an number ex(n,H)
is the maximum number of edges in a k-graph on n vertices that does not contain H as a subgraph;
∗UniversityofSouthCarolina,Columbia,SC29208, ([email protected]).
†UniversityofSouthCarolina,Columbia,SC29208,([email protected]). ThisauthorwassupportedinpartbyNSF
grantDMS1000475.
1
such a graph is called H-free. Katona et al. [24] showed that f(n,H)=ex(n,H)/ n is a decreasing
k
function of n. The limit π(H)= lim f(n,H), which always exists, is called the Tur´an density of H.
n (cid:0) (cid:1)
For k = 2, the graph case, Erd→o˝∞s-Stone-Simonovits proved π(G) = 1 1 for any graph G
− χ(G) 1
withchromaticnumberχ(G) 3. IfGisbipartite,thenex(n,G)=o(n2). Them−agnitudeofex(n,G)
≥
is unknown for most bipartite graphs G.
LetKr denotethecompleter-graphonk vertices. Tur´andeterminedthevalueofex(n,K2)which
k k
implyies that π(K2) = 1 1 for all k 3. However, no Tur´an density π(Kr) is known for any
k − k 1 ≥ k
k > r 3. The most extens−ively studied case is when k = 4 and r = 3. Tur´an conjectured [37]
≥
that π(K3)= 5/9. Erdo˝s [13] offered $500 for determining any π(Kr) with k > r 3 and $1000 for
4 k ≥
answeringitforallkandr. Theupperboundsforπ(K3)havebeensequentiallyimproved: 0.6213(de
4
Caen[6]),0.5936(Chung-Lu[7]),0.56167(Razborov[35],usingflagalgebramethod.) Thereareafew
uniform hypergraphs whose Tur´an density has been determined: the Fano plane [16, 27], expanded
triangles[28], 3-books,4-books[17], F [15], extended complete graphs[34], etc. In particular,Baber
5
[2] recently found the Tur´an density of many 3-uniform hypergraphs using flag algebra methods. For
a more complete survey of methods and results on uniform hypergraphs see Peter Keevash’s survey
paper [26].
Anon-uniformhypergraphH =(V,E)consistsofavertexsetV andanedgesetE 2V. Herethe
⊆
edges of E could have different cardinalities. The set of all the cardinalities of edges in H is denoted
by R(H), the set of edge types. For a fixed hypergraph H, the Tur´an density π(H) is defined to be
lim max h (G ), where the maximum is taken over all H-free hypergraphs G on n vertices
n→∞ Gn n n n
satisfying R(G ) R(H). h (G ), the so called Lubell function, is the expected number of edges
n n n
⊆
in G hit by a random full chain. The Lubell function has been a very useful tool in extremal poset
n
theory, in particular it has been used in the study of the diamond conjecture.
In section 2, we show that our notion of π(H) is well-defined and is consistent with the usual
definition for uniform hypergraphs. We also give examples ofTur´andensities for severalnon-uniform
hypergraphs. In section 3, we generalize the supersaturation Lemma to non-uniform hypergraphs.
Then we prove that blowing-up will not affect the Tur´an density. Using various techniques, we
determinetheTur´andensityofevery 1,2 -hypergraphinsection4. Remarkably,theTur´andensities
{ }
of 1,2 -hypergraphsare in the set
{ }
9 5 3 5 1
1, , , , ,...,2 ,... .
8 4 2 3 − k
(cid:26) (cid:27)
Among r-uniform hypergraphs, r-partite hypergraphs have the smallest possible Tur´an density.
Erdo˝sprovedthatanyr-uniformhypergraphforbiddingthecompleter-uniformr-partitehypergraphs
can have at most O(nr 1/δ) edges. We generalize this theorem to non-uniform hypergraphs. A
−
hypergraph is degenerate if it has the smallest possible Tur´an density. For r-uniform hypergraphs, a
hypergraphH is degenerateif andonlyif H is the subgraphofa blow-upofa single edge. Unlike the
degenerate r-uniform hypergraphs, the degenerate non-uniform hypergraphs are not classified. For
non-uniformhypergraphs,chains–onenaturalextensionofasingleedge–aredegenerate. Additionally,
everysubgraphofablow-upofachainisalsodegenerate. However,wegiveanexampleofadegenerate,
non-uniform hypergraph not contained in any blow-up of a chain. This leaves open the question of
which non-uniform hypergraphs are degenerate.
In section 6, we consider the suspension of hypergraphs. The suspension of a hypergraph H is a
new hypergraph, denoted by S(H), with one additional vertex, , added to every edge of H. In a
∗
2
hypergraph Tur´an problem workshop hosted by the AIM Research Conference Center in 2011, the
followingconjecturewasposed: lim π(St(Kr))=0. Weconjecture lim π(St(H))= R(H) 1holds
t n t | |−
for any hypergraph H. Some par→ti∞al results are proved. Finally in th→e∞last section, we will point out
the relation between the Tur´an problems on hypergraphs and extremal poset problems.
2 Non-uniform hypergraphs
2.1 Notation
Recall that a hypergraph H is a pair (V,E) with the vertex set V and edge set E 2V. Here we
⊆
place no restriction on the cardinalities of edges. The set R(H) := F : F E is called the set of
{| | ∈ }
its edge types. A hypergraph H is k-uniform if R(H) = k . It is non-uniform if it has at least two
{ }
edge types. For any k R(H), the level hypergraph Hk is the hypergraph consisting of all k-edges
∈
of H. A uniform hypergraph H has only one (non-empty) level graph, i.e., H itself. In general, a
non-uniform hypergraph H has R(H) (non-empty) level hypergraphs. Throughout the paper, for
| |
any finite set R of non-negative integers, we say, G is an R-graph if R(G) R. We write GR for a
⊆ n
hypergraph on n vertices with R(G) R. We simplify it to G if n and R are clear under context.
⊆
Let R be a fixed set ofedge types. Let H be an R-graph. The number of vertices in H is denoted
by v(H) := V(H). Our goal is to measure the edge density of H and be able to compare it (in a
| |
meaningful way) to the edge density of other R graphs. The standard way to measure this density
would be:
E(H)
µ(H)= | | .
v(H)
k R(H) k
∈
This density ranges from 0 to 1 (as one woulPd expect)(cid:0)–a co(cid:1)mplete R-graph having a density of 1.
Unfortunately, this is no longera useful measureofdensity since the number of edgeswith maximum
cardinality will dwarf the number of edges of all other sizes. Specifically, one could take k-uniform
hypergraph (where k = max r : r R(H) ) on enough vertices and make its density as close to 1
{ ∈ }
he likes. The problemis that this k-uniformhypergraphis quite different fromthe complete R-graph
(when R >1)with the same number ofvertices. Instead, we use the Lubell function to measurethe
| |
edge density. This is adapted from the use of the Lubell function studying families of subsets.
For a non-uniform hypergraph G on n vertices, we define the Lubell function of G as
1 E(Hk)
h (G):= = | |. (1)
n n n
F∈XE(G) |F| k∈XR(G) k
(cid:0) (cid:1) (cid:0) (cid:1)
The Lubell function is the expected number of edges hit by a random full chain. Namely, pick a
uniformly random permutation σ on n vertices; define a random full chain C by
σ
, σ(1) , σ(1),σ(2) , ,[n] .
{{∅} { } { } ··· }
Let X = E(G) C , the number of edges hit by the random full chain. Then
σ
| ∩ |
h (G)=E(X). (2)
n
Given two hypergraphsH and H , we say H is a subgraphof H , denoted by H H , if there
1 2 1 2 1 2
⊆
existsa1-1mapf: V(H ) V(H )sothatf(F) E(H )foranyF E(H ). Wheneverthisoccurs,
1 2 2 1
→ ∈ ∈
3
f
we say the image f(H ) is an ordered copy of H , written as H ֒ H . A necessary condition for
1 2 1 2
→
H H isR(H ) R(H ). GivenasubsetK V(H)andasubsetS R(H),theinducedsubgraph,
1 2 1 2
⊆ ⊆ ⊆ ⊆
denotedbyH[S][K],isahypergraphonK withtheedgeset F E(H): F K and F S . When
{ ∈ ⊆ | |∈ }
S =R(H), we simply write H[K] for H[S][K].
Given a positive integer n and a subset R [n], the complete hypergraphKR is a hypergraphon
⊆ n
n vertices with edge set i R [ni] . For example, Kn{k} is the complete k-uniform hypergraph. Kn[k]
is the non-uniform hypergra∈ph with all possible edges of cardinality at most k.
S (cid:0) (cid:1)
5 4
2
6 3
1
3 1 2
Illustration of K3{2,3} A (tight) cycle C6{2,3}
Given a family of hypergraphs with common set of edge-types R, we define
H
π ( ):=max h (G): v(G)=n,G KR, and G contains no subgraph in .
n H n ⊆ n H
A hypergraph G:=G(cid:8)R is extremal with respect to the family if (cid:9)
n H
1. G contains no subgraph in .
H
2. h (G)=π ( ).
n n
H
The Tur´an density of is defined to be
H
π( ):= lim π ( )
n
H n H
→∞
1
= lim max : v(G)=n,G KR, and G contains no subgraph in
n  n ⊆ n H
→∞ F∈XE(G) |F| 
(cid:0) (cid:1)
whenthelimitexists;wewillsoonshowthatthislimitalwaysexists. When containsonehypergraph
H
H, then we write π(H) instead of π( H ).
{ }
Throughout, we will consider n growing to infinity, and R to be a fixed set (not growing with n).
Note that π( ) agrees with the usual definition of
H
ex( ,n)
π( )= lim H
H n n
→∞ k
(cid:0) (cid:1)
4
when is a set of k-uniform hypergraphs. The following result is a direct generalization Katona-
H
Nemetz-Simonovit’s theorem [24].
Theorem 1. For any family of hypergraphs with a common edge-type R, π( ) is well-defined, i.e.
H H
the limit lim π ( ) exists.
n
n H
→∞
Proof. It suffices to show that π ( ), viewed as a sequence in n, is decreasing.
n
H
Write R = k ,k ,...,k . Let G KR be a hypergraph with v(G )= n not containing and
{ 1 2 r} n ⊆ n n H
with Lubell value h (G ) = π ( ). For any ℓ < n, consider a random subset S of the vertices of G
n n n
H
with size S =ℓ.
| |
Let G [S] be the induced subgraph of G (whose vertex set is restricted to S). Clearly
n n
π ( ) E(h (G [S])).
ℓ ℓ n
H ≥
Write E(G ) = E E ... E where E contains all the edges of size k . Note that the
n k1 k2 kr ki i
(ℓ)
expected number of eSdges ofSsizeSki in Gn[S] is precisely (kni)|Eki|. It follows that
ki
π ( ) E(h (G [S]))
ℓ ℓ n
H ≥
r E(E S )
= | ki ki |
ℓ
Xi=1 kTi (cid:0) (cid:1)
r ((knℓi))|E(cid:0)ki|(cid:1)
= ki
ℓ
Xi=1 ki
r
= |Ek(cid:0)i|(cid:1)
n
Xi=1 ki
=hn(G(cid:0)n)(cid:1)
=π ( ).
n
H
The sequence π ( ) is non-negative and decreasing; therefore it converges.
n
H
For afixedsetR:= k ,k ,...,k (withk <k < <k ), anR-flagis anR-graphcontaining
1 2 r 1 2 r
{ } ···
exactly one edge of each size. The chain CR is a special R-flag with the edge set
E(CR)= [k ],[k ],...,[k ] .
1 2 r
{ }
Proposition 1. For any hypergraph H, the following statements hold.
1. R(H) 1 π (H) R(H).
n
| |− ≤ ≤| |
2. For subgraph H of H, we have π(H ) π(H) R(H) + R(H ).
′ ′ ′
≤ −| | | |
3. For any R-flag L on m vertices and any n m, we have π (L)= R 1.
n
≥ | |−
5
Proof: Pick any maximal proper subset R of R(H). Consider the complete graph KR′. Since
′ n
KR′ misses one type of edge in R(H) R, it does not contain H as a subgraph. Thus
n \ ′
π (H) h (KR′)= R = R(H) 1.
n ≥ n n | ′| | |−
The upper bound is due to the fact h (KR(H))= R(H).
n n
| |
Proofofitem 2 is similar. LetS =R(H ) andGS be anextremalhypergraphfor π (H ). Extend
′ n n ′
GS to GR(H) by adding all the edges with cardinalities in R(H) S. The resulting graph GR(H) is
n n \ n
H-free. We have
πn(H)≥πn(GnR(H))=πn(GSn)+|R(H)|−|S|=|R(H)|−|R(H′)|+πn(H′).
Taking the limit as n goes to infinity, we have
π(H) R(H) R(H ) +π(H ).
′ ′
≥| |−| |
Finally, for item 3, consider L-free hypergraph GR. Pick a random n-permutation σ uniformly.
n
LetX bethenumberofedgesofGR hitbyarandomflagσ(L). NotethateachedgeF hasprobability
n
1 of being hit by σ(L). We have
(n)
|F|
1
E(X)= =h (G). (3)
n n
F∈XE(G) |F|
(cid:0) (cid:1)
Since GR is L-free, we have X r 1. Taking the expectation, we have
n ≤ −
h (GR)=E(X) r 1.
n n ≤ −
Hence, π (H) r 1. The result is followed after combining with item 1. (cid:3)
n
≤ −
Definition 1. A hypergraph H is called degenerate if π(H)= R(H) 1.
| |−
ByProposition1,flags,andspecificallychains,aredegeneratehypergraphs. Anecessarycondition
for H to be degenerate is that every level hypergraph Hki is ki-partite. The following examples will
show that the converse is not true.
Example 1: The complete hypergraph K2{1,2} has three edges {1}, {2}, and {1,2}. We claim
5
π(K2{1,2})= 4. (4)
The lower bound is from the following construction. Partition [n] into two parts A and B of nearly
equal size. Consider the hypergraph G with the edge set
A [n] A
E(G)= .
1 ∪ 2 \ 2
(cid:18) (cid:19) (cid:18)(cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
It is easy to check hn(G)= 45 +O(n1) and that G contains no copy of K2{1,2}.
6
Now we prove the upper bound. Consider any K2{1,2}-free hypergraph G of edge-type {1,2} on n
vertices. Let A be the set of all singleton edges. For any x,y A, xy is not a 2-edge of G. We have
∈
h (G) |A| +1 |A2|
n ≤ n − n
(cid:0) 2 (cid:1)
A A2 1
(cid:0) (cid:1)
=1+ | | | | +O
n − n2 n
(cid:18) (cid:19)
1 1
1+ +O .
≤ 4 n
(cid:18) (cid:19)
In the last step, we use the fact that f(x) = 1+x x2 has the maximum value 5. Combining the
− 4
upper and lower bounds we have π(K2{1,2})= 45.
1,k
The argument is easily generalized to the complete graph K{ } (for k>1). We have
k
k 1
π(K1,k)=1+ − . (5)
k k
kk−1
Definition 2. Let H be a hypergraph. The suspension of H, denoted by S(H), is a new hypergraph
with the vertex set V(H) and the edge set F : F E(H) . Here is a new vertex not in
∪{∗} { ∪{∗} ∈ } ∗
H.
Definition 3. Let H be a hypergraph. The k-degree of a vertex x, denoted d (x), is the number of
k
edges of size k containing x.
1,2
Example 2: Consider H :=S(K{ }). The edges of H are 1,2 , 2,3 , 1,2,3 . We claim
2 { } { } { }
5
π(S(K2{1,2}))= 4.
Thelowerboundisfromthefollowingconstruction. Partition[n]intotwopartsAandB ofnearly
equal size. Consider the hypergraph G with the edge set E = E E where E = A B and
2 3 2 2 2
E3 = [n3] \ A3 B3 . It is easy to check hn(G)= 54 +O n1 anSd that G is H-free.(cid:0) (cid:1)S(cid:0) (cid:1)
No(cid:0)w w(cid:1)e(cid:16)p(cid:0)rov(cid:1)eSt(cid:0)he(cid:1)u(cid:17)pper bound. Consider any H-free h(cid:0)yp(cid:1)ergraph G of edge-type {2,3} on n
vertices. Recall that d (v) denotes the number of 2-edges that contain v. For each pair of 2-edges
2
that intersect v there is a unique 3-set containing those two pairs. This 3-set cannot appear in the
edge set of G since G is H-free. We say that the edge is forbidden. Note that each 3-edge may be
forbidden up to 3 times in this manner–depending on which of the three vertices we call v. Hence
there are at least 1 d2(v) 3-edges which are not in G. Note that this is true for any H-free
3 v [n] 2
graph G with number o∈f vertices. Hence
P (cid:0) (cid:1)
n 1 d2(v) 1 d (v)
hn(G)≤ 3 − 3 nv∈[n] 2 + 2 v∈[nn] 2
(cid:0) (cid:1) P3 (cid:0) (cid:1) P 2
v [n(cid:0)]d(cid:1)2(v)2 1 (cid:0)1(cid:1)
=1− P ∈6 n3 + 6 n3 + 2 n2 !vX∈[n]d2(v)+1.
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
7
Applying Cauchy-Schwarz Inequality and letting m:= d (v), we have
v 2
2 P
d (v)
− v [n] 2 1 1
h (G) ∈ + + d (v)+1
n ≤ (cid:16)P6n n (cid:17) 6 n 2 n ! 2
3 3 2 vX∈[n]
m2 m(cid:0) (cid:1) 1 (cid:0) (cid:1) (cid:0) (cid:1)
= + +1+O
−n4 n2 n
(cid:18) (cid:19)
5 1
+O .
≤ 4 n
(cid:18) (cid:19)
In the last step, we use the fact that f(x)=1+x x2 has the maximum value 5. Taking the limit,
− 4
we get π(S(K2{1,2}))≤ 45.
We can generalize this construction, giving the following lower bound for Sk(K2{1,2}) (the k-th
suspension of K2{1,2}). The details of the computation are omitted.
1
π(Sk(K2{1,2}))≥1+ 2k+1. (6)
Conjecture 1. For any k ≥2, π(Sk(K2{1,2}))=1+ 2k1+1.
3 Supersaturation and Blowing-up
SupersaturationLemma [14] is an important tool for uniform hypergraphs. There is a naturalgener-
alization of the supersaturation lemma and blowing-up in non-uniform hypergraphs.
Lemma 1. (Supersaturation) For any hypergraph H and a>0 there are b, n >0 so that if G is
0
a hypergraph on n >n vertices with R(G)=R(H) and h (G) >π(H)+a then G contains at least
0 n
b n copies of H.
v(H)
(cid:0) Pr(cid:1)oof: LetR:=R(H)andr := R. Sinceπ(H)= lim π (H),thereisann sothatform n ,
n 0 0
| | n ≥
π (H) π(H)+ a. For any n m n, there must b→e ∞at least a n m-sets M V(G) inducing
m ≤ 2 0 ≤ ≤ 2r m ⊂
a R-graph G[M] with h(G[M])>π(H)+ a. Otherwise, we have
2 (cid:0) (cid:1)
a n a n n
h (G[M]) π(H)+ + r =(π(H)+a) .
m
≤ 2 m 2r m m
XM (cid:16) (cid:17)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
8
But we also have
1
h (G[M])=
m m
XM XM F∈XE(G) |F|
F M
⊆ (cid:0) (cid:1)
1
=
m
F∈XE(G)MX⊇F |F|
n F(cid:0) (cid:1)
−| |
m F
= −| |
m
(cid:0) (cid:1)
F∈XE(G) |F|
(cid:0)n (cid:1)
= m
n
F∈XE(G) (cid:0)|F|(cid:1)
n (cid:0) (cid:1)
= h (G).
n
m
(cid:18) (cid:19)
This is a contradiction to the assumption that h (G)>π(H)+a. By the choice of m, each of these
n
m-sets containsa copy ofH, so the number of copies ofH inG is atleast a n / n−v(H) =b n
2r m m v(H) v(H)
where b:= a m −1. (cid:0) (cid:1) (cid:0) − (cid:1) (cid:0) (cid:3)(cid:1)
2r v(H)
Supersaturation can be used to show that “blowing up” does not change the Tur´an density π(H)
(cid:0) (cid:1)
just like in the uniform cases.
Definition 4. For any hypergraph H and positive integers s ,s ,...,s , the blowup of H is a new
n 1 2 n
hypergraph (V,E), denoted by H (s ,s ,...,s ), satisfying
n 1 2 n
1. V := n V , where V =s .
⊔i=1 i | i| i
2. E = V .
∪F∈E(H) i∈F i
When s =s = Q=s =s, we simply write it as H(s).
1 2 n
···
Considerthe followingsimple example. TakeH to be the hypergraphwithvertex set[3]andedge
set E = 1,2 , 1,2,3 ,a chain. Consider the blow-ups H(2,1,1) and H(1,1,2) illustrated below.
{{ } { }}
v v
1 1,1 3,1
v
1
v v v
3 2 3 2
2
v v
1,2 3,2
H H(2,1,1) H(1,1,2)
In the blow-up H(2,1,1) vertex 1 splits into vertices v and v ; vertex 2 becomes v and vertex 3
1,1 1,2 2
becomes v . In the blow-up H(1,1,2) vertex 3 splits into vertices v and v ; vertex 1 becomes v
3 3,1 3,2 1
and vertex 2 becomes v .
2
9
Theorem 2. (Blowing up) Let H be any hypergraph and let s 2. Then π(H(s))=π(H).
≥
Proof: Let R:=R(H). By the supersaturation lemma, for any a>0 there is a b>0 and an n
0
so that any R-graph G on n n vertices with h (G)>π(H)+a contains at least b n copies of
≥ 0 n v(H)
H. Consider an auxiliary v(H)-uniform hypergraphU on the same vertex set as G where edges of U
(cid:0) (cid:1)
correspond to copies of H in G. For any S >0, there is a copy of K =Kv(H)(S) in U. This follows
v(H)
fromthe factthat π(Kv(H)(S))=0 since it is v(H)-partite,andh (U)=b>0. Now coloreachedge
v(H) n
of K by one of v(H)! colors, each color corresponding to one of v(H)! possible orders the vertices
of H are mapped to the parts of K. The pigeon-hole principle gives us that one of the color classes
contains at least Sv(H)/v(H)! edges. For large enough S there is a monochromatic copy of Kv(H)(s),
v(H)
which gives a copy of H(s) in G. (cid:3)
Corollary 1. (Squeeze Theorem) Let H be any hypergraph. If there exists a hypergraph H and
′
integer s 2 such that H H H (s) then π(H)=π(H ).
′ ′ ′
≥ ⊆ ⊆
Proof: One needs only observe that for any hypergraphs H H H that π(H ) π(H )
1 2 3 1 2
π(H ). If H =H (s) for some s 2 then π(H )=π(H ) by the⊆previo⊆us theorem. ≤ ≤(cid:3)
3 3 1 1 3
≥
4 Tur´an Densities of 1,2 -hypergraphs
{ }
In this section we will determine the Tur´an density for any hypergraph H with R(H) = 1,2 . We
{ }
begin with the following more general result.
Theorem 3. Let H =H1 Hk be a hypergraph with R(H)= 1,k and E(H1)=V(Hk). Then
∪ { }
1+π(Hk) if π(Hk) 1 1;
π(H)= 1/(k 1) ≥ − k
1+ 1 − 1 1 otherwise.
 k(1 π(Hk)) − k
−
(cid:16) (cid:17) (cid:0) (cid:1)
Proof: For each n N,let G be any H-free graph n vertices with h (G ) = π (H). Partition
n n n n
the vertices of G into∈X = v V(G ) : v E(G) and X¯ containing everything else. Say
n n n n
that X = x n and X¯ ={(1∈ x )n for{so}m∈e x }[0,1]. Since (x ) is a sequence in [0,1]
n n n n n n
| | | | − ∈
it has a convergent subsequence. Consider (x ) to be the convergent subsequence, and say that
n
x x [0,1]. With the benefit of hindsight, we know that x > 0, however, for the upper bound
n
→ ∈
portion of this proof we will not assume this knowledge.
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