We obtain a general bound on the Turán density of a hypergraph in terms of the number of edges that it contains. If F is an r-uniform hypergraph with f edges

we show that [pi](F) =3 and f->[infinity]

Keevash, Peter

English

Caltech Authors - Main

The Turan problem for hypergraphs of xed size
Peter Keevash
Department of Mathematics
Caltech, Pasadena, CA 91125, USA.
keevash@caltech.edu
Submitted: Oct 22, 2004; Accepted: Jun 3, 2005; Published: Jun 14, 2005
Mathematics Subject Classications: 05D05
Abstract
We obtain a general bound on the Turan density of a hypergraph in terms of
the number of edges that it contains. If F is an r-uniform hypergraph with f edges
we show that (F) < f−2f−1 − (1 + o(1))(2r!2=rf3−2=r)−1, for xed r  3 and f !1.
Given an r-uniform hypergraph F , the Turan number of F is the maximum number
of edges in an r-uniform hypergraph on n vertices that does not contain a copy of F .
We denote this number by ex(n;F). It is not hard to show that the limit (F) =
limn!1 ex(n;F)=
(
n
r

exists. It is usually called the Turan density of F . There are very
few hypergraphs with r > 2 for which the Turan density is known, and even fewer for the
exact Turan number. We refer the reader to [10, 11, 12, 13, 14, 15, 16] for recent results
on these problems.
A general upper bound on Turan densities was obtained by de Caen [3], who showed
(K
(r)
s )  1 −
(
s−1
r−1
−1
, where K
(r)
s denotes the complete r-uniform hypergraph on s
vertices. A construction showing (K
(r)
s )  1−
(
r−1
s−1
r−1
was given by Sidorenko [17] (see
also [18]); better bounds are known for large r. We refer the reader to Sidorenko [18] for a
full discussion of this problem. For a general hypergraph F Sidorenko [19] (see also [20])
obtained a bound for the Turan density in terms of the number of edges, showing that if
F has f edges then (F)  f−2
f−1 . In this note we improve this as follows.
Theorem 1 Suppose F is an r-uniform hypergraph with f edges.
(i) If r = 3 and f  4 then (F)  1
2
(
p
f 2 − 2f − 3− f + 3).
(ii) For a xed r  3 and f !1 we have (F) < f−2
f−1 − (1 + o(1))(2r!2=rf 3−2=r)−1.
We start by describing our main tool, which is Sidorenko’s analytic approach. See [20]
for a survey of this method. Consider an r-uniform hypergraph H on n vertices. It is
convenient to regard the vertex set V as a nite measure space, in which each vertex v has
(fvg) = 1=n, so that (V ) = 1. We write h : V r ! f0; 1g for the symmetric function
the electronic journal of combinatorics 12 (2005), #N11 1
h(x1;    ; xr) which takes the value 1 if fx1;    ; xrg is an edge of H and 0 otherwise.
Then
R
h dr = r!e(H)n−r = d + O(1=n), where d = (n
r
−1
e(H) is the density of H.
Now consider a xed forbidden r-uniform hypergraph F with f edges on the vertex
set f1;    ; mg. We associate to vertex i the variable xi, and to an edge e = fi1;    ; irg
the function he(x) = h(xi1 ;    ; xir), where x denotes the vector (x1;    ; xm). The con-
guration product of F with respect to h is the function hF (x) =
Q
e2F he(x). ThenZ
hFdm = n−mhom(F ;H) = n−mmon(F ;H)+O(n−1) = n−maut(F)sub(F ;H)+O(n−1);
where hom(F ;H) is the number of homomorphisms (edge-preserving maps) from F to H,
mon(F ;H) is the number of these that are monomorphisms (injective homomorphisms),
aut(F) is the number of automorphisms of F and sub(F ;H) is the number of F -subgraphs
of H. Also, Erd}os-Simonovits supersaturation [6] implies that for any  > 0 there is  > 0
and an integer n0 so that for any r-uniform hypergraph H on n  n0 vertices with(
n
r
−1
e(H) > (F) +  we have n−msub(F ;H) > . It follows that
(F) = inf
>0
lim inf
jV j!1
max
h:V r!f0;1g; R hF dm<
Z
h dr: (1)
We say that F is a forest if we can order its edges as e1;    ; ef so that for every
2  i  f there is some 1  j  i − 1 so that ei \
([i−1t=1et  ej . Sidorenko [20] showed
that if F is a forest with f edges then
Z
hF dm 
Z
h dr
f
: (2)
Now we need a lemma on when a hypergraph contains a forest of given size.
Lemma 2 (i) An r-uniform hypergraph with at least r!(t − 1)r edges contains a forest
with t edges.
(ii) Let F be a 3-uniform hypergraph. Then either (a) F contains a forest with 3 edges,
or (b) (F) = 0, or (c) F  K(3)4 , or (d) F = F5 = fabc; abd; cdeg.
Proof. (i) This is immediate from the result of Erd}os and Rado [5] that such a hyper-
graph contains a sunflower with t petals, i.e. edges e1;    ; et for which all the pairwise
intersections ei \ ej are equal. A sunflower is in particular a forest.
(ii) Consider a 3-uniform hypergraph F that does not contain a forest with 3 edges.
We can assume that F is not 3-partite (Erd}os [4] showed that this implies (F) = 0) so
F has at least 3 edges. Clearly F cannot have two disjoint edges, as then adding any
other edge gives a forest.
Suppose there is a pair of edges that share two points, say e1 = abc and e2 = abd. Any
other edge must contain c and d, or together with e1 and e2 we have a forest. Consider
another edge e3 = cde. If there are no other edges then either F = F5 or F  K(3)4 (if e
the electronic journal of combinatorics 12 (2005), #N11 2
equals a or b). If there is another edge e4 = cdf then the same argument shows that e1
and e2 both contain e and f , i.e. F = K(3)4 and there can be no more edges.
The other possibility is that every pair of edges intersect in exactly one point. Then
there are at most 2 edges containing any point, or we would have a forest with 3 edges.
Consider three edges, which must have the form e1 = abc, e2 = cde, e3 = efa. There can
be at most one more edge e4 = bdf . But this forms a 3-partite hypergraph (with parts
ad, be, cf), a case we have already excluded. This proves the lemma. 
Proof of Theorem. Let F be an r-uniform hypergraph with f edges that contains a
forest T with t edges. Label the edges e1;    ; ef , where e1;    ; et are the edges of T .
Suppose that H is an r-uniform hypergraph on a vertex set V of size n. Dene the
measure  and the function h : V r ! f0; 1g as before. Observe the inequality
hF(x)  hT (x) +
fX
i=t+1
he1(x)(hei(x)− 1):
This holds, as the second term is non-positive (since he(x) 2 f0; 1g), so it could only fail
for some x if hF(x) = 0 and hT (x) = 1. But then we have he1(x) =    = het(x) = 1 and
hei(x) = 0 for some i > t, and the term he1(x)(hei(x) − 1) = −1 cancels hT (x), so the
inequality holds for all x. Integrating gives
Z
hF (x) dm 
Z
hT (x) dm +
fX
i=t+1
Z
he1(x)hei(x)− he1(x) dm  pt + (f − t)(p2 − p);
where we write p =
R
h dr and apply the inequality (2) for the forests T and fe1; eig,
t + 1  i  f . By equation (1) we deduce that the Turan density  = (F) satises
t + (f − t)(2 − )  0.
Writing g(x) = xt−1 + (f − t)(x − 1) we either have  = 0 or g()  0. Now
g(0) = −(f − t)  0, g(1) = 1 and dg
dx
= (t − 1)xt−2 + f − t  0 for 0 < x < 1 so g has
exactly one root  in [0; 1], and   .
First we consider the case r = 3. If f  5 then by the lemma we can take t = 3. Solving
the quadratic g(x) = x2+(f−3)(x−1) = 0 gives    = 1
2
(
p
f 2 − 2f − 3−f +3). This
also holds when f = 4, as then by the lemma we may suppose that F = K(3)4 . Chung and
Lu [2] showed that (K
(3)
4 )  3+
p
17
12
which is less than 1
2
(
p
5− 1).
Now consider the case when r  3 is xed and f !1. By the lemma we can take t =
(f=r!)1=r. Write  = 1−. Since g() = 0 we have (f−t) = (1−)t−1 < 1, so  < 1=(f−t).
From the Taylor expansion of (1− )t−1 we have (f − t) > 1− (t−1)+ (t−1
2

2− (t−1
3

3.
Also
(
t−1
3

3 < 1
6

t−1
f−t
3
< 1
6
(t=f)3 (since f > t2) so
(
t−1
2

2 − (f − 1) + 1− 1
6
(t=f)3 < 0.
the electronic journal of combinatorics 12 (2005), #N11 3
Writing  = (f − 1)2− 4(t−1
2

(1− 1
6
(t=f)3) for the discriminant of this quadratic we have
 >
f − 1−1=2
(t− 1)(t− 2) =
2(1− 1
6
(t=f)3)
f − 1 + 1=2
=
2
f − 1
 
1 +

1− 2(t− 1)(t− 2)(1− 1
6
(t=f)3)(f − 1)−2
1=2!−1
+ O(t3=f 4)
=
2
f − 1
(
1 + 1− (t− 1)(t− 2)(f − 1)−2 + O(t4=f 4)−1 + O(t3=f 4)
=
1
f − 1(1 +
1
2
(t− 1)(t− 2)(f − 1)−2 + O(t4=f 4)) + O(t3=f 4)
=
1
f − 1 +
(t− 1)(t− 2)
2(f − 1)3 + O(t
3=f 4):
Since  = 1−  and t = (f=r!)1=r we have
   < f − 2
f − 1 − (1 + o(1))(2r!
2=rf 3−2=r)−1:
This proves the theorem. 
Remarks. (1) For a graph G we have e(G)  ((G)
2

with equality if and only if G is
complete. The Erd}os-Stone theorem [7] implies that (G) = (G)−2
(G)−1 < 1 − 1+o(1)p2e(G) . It is
natural to think that complete hypergraphs should also have the highest Turan density
among all hypergraphs with the same number of edges. Were this true de Caen’s bound
would give (F) < 1− Ω(f−(r−1)=r) for an r-uniform hypergraph F with f edges.
(2) If F has 3 edges then Sidorenko’s bound (F)  1=2 is tight when F = K(2)3
is a triangle, or more generally when F is the 2k-uniform hypergraph with edges fP1 [
P2; P2 [ P3; P3 [ P1g, where P1; P2; P3 are disjoint sets of size k (see [8, 14]). If F is
3-uniform and has 3 edges then the lemma shows that (F)  maxf(F4); (F5)g, where
F4 denotes the 3-edge subgraph of K(3)4 and F5 = fabc; abd; cdeg. Frankl and Fu¨redi
[9] showed that (F5) = 2=9 and Mubayi [15] showed (F4) < 1=3 − 10−6, so we see
that (F) < 1=3 − 10−6, and Sidorenko’s bound is not tight. It would be interesting to
determine if it is ever tight for a hypergraph with edges of odd size.
(3) How many edges in an r-uniform hypergraph guarantee a forest with t edges? An
answer to this question may lead to an improvement in our theorem, and it also seems
interesting in its own right. Erd}os and Rado [5] conjectured that for any t there is a
constant C so that any r-uniform hypergraph with Cr edges contains a sunflower with t
edges. We can obtain a bound of this form for forests, indeed, we claim that any r-uniform
hypergraph F with (2t)r edges contains a forest with t edges. For if we x any edge e,
then the other edges have 2r possible intersections with it, so we can nd a hypergraph
F 0  Fne with (2t−1)r edges, all of which have the same intersection with e. By induction
we can nd a forest with t− 1 edges in F 0, and adding e gives a forest of size t in F .
the electronic journal of combinatorics 12 (2005), #N11 4
Actually, it is not hard to improve this bound to 2
(
r
r=2
t−2
. For we only need the
intersections fe\e0 : e0 2 Fg to form a chain, and the subsets of e can be partitioned into(
r
r=2

chains (see, for example, [1] page 10). Thus we need only lose a factor
(
r
r=2

at each
induction step, and after t− 2 steps we get down to a 2-edge forest.
However, this bound does not help in our application, as we are interested in the case
when r is xed and t is large. We have an upper bound of r!tr from Erd}os and Rado, and
and noting that K
(r)
r+t−2 does not contain a forest with t edges we obtain a lower bound
of
(
r+t−2
r
  tr=r!, so we have a constant r!2 factor of uncertainty.
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the electronic journal of combinatorics 12 (2005), #N11 6


The Turán problem for hypergraphs of fixed size

A new generalization of the Erd}os-Ko-Rado theorem,

A. F. Sidorenko, Extremal combinatorial problems in spaces with continuous measure, Issled. Operatsi˘ıi ASU 34 (1989), 34{40.

A. F. Sidorenko, Systems of sets that have the T -property, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1981, 19{22.

A. F. Sidorenko, What we know and what we do not know about Turan numbers, Graphs Combin. 11 (1995), 179{199.

An analytic approach to extremal problems for graphs and hypergraphs, Extremal problems for  sets (Visegrad,

An upper bound for the Turan number t (n, 4),

Asymptotic solution of a Turan-type problem.

B. Bollobas, Combinatorics, Set systems, hypergraphs, families of vectors and combinatorial probability, Cambridge University Press, Cambridge, 1986.

Combinatorics, Set systems, hypergraphs, families of vectors and combinatorial probability,

D. de Caen, Extension of a theorem of Moon and Moser on complete subgraphs, Ars Combin. 16 (1983), 5{10.

D. Mubayi and V. Ro¨dl, On the Turan number of Triple Systems, J. Comb. Theory Ser. A, 100 (2002), 136{152.

D. Mubayi, On hypergraphs with every four points spanning at most two triples, Electron. J. Combin. 10 (2003), Note 10, 4 pp. (electronic).

Extension of a theorem of Moon and Moser on complete subgraphs,

Extremal combinatorial problems in spaces with continuous measure,

F. Chung and L. Lu, An upper bound for the Turan number t (n, 4), J. Combin. Theory Ser. A 87 (1999), 381{389.

Intersection theorems for systems of sets,

On a hypergraph Turan problem of Frankl, Combinatorica,

On extremal problems of graphs and generalized graphs,

On hypergraphs with every four points spanning at most two triples,

On the structure of linear graphs,

On the Turan number of Triple Systems,

On triple systems with independent neighborhoods,

P. Erd}os and A. H. Stone, On the structure of linear graphs, Bull. Amer. Math. Soc. 52 (1946), 1087{1091.

P. Erd}os and M. Simonovits, Supersaturated graphs and hypergraphs, Combinatorica 3 (1983), 181{192.

P. Erd}os and R. Rado, Intersection theorems for systems of sets, J. London Math. Soc. 35 1960, 85{90.

P. Erd}os, On extremal problems of graphs and generalized graphs, Israel J. Math. 2 (1964), 183{190.

P. Frankl, Asymptotic solution of a Turan-type problem. Graphs and Combinatorics 6 (1990), 223{227.

P. Frankl, Z. Fu¨redi, A new generalization of the Erd}os-Ko-Rado theorem, Combinatorica 3 (1983), 341{349.

P. Keevash and B. Sudakov, On a hypergraph Turan problem of Frankl, Combinatorica, to appear.

P. Keevash and B. Sudakov, The exact Turan number of the Fano plane, Combinatorica, to appear.

P. Keevash, The Turan problem for projective geometries, J. Combin. Theory Ser. A, to appear. the electronic journal of combinatorics 12 (2005), #N11 5

Supersaturated graphs and hypergraphs,

Systems of sets that have the T -property,

The exact Turan number of the Fano plane, Combinatorica,

The Turan problem for projective geometries,

Triple systems not containing a Fano Con

What we know and what we do not know about Turan numbers,

Z. Fu¨redi and M. Simonovits, Triple systems not containing a Fano Con Combin. Probab. Comput., to appear.

Z. Fu¨redi, O. Pikhurko and M. Simonovits, On triple systems with independent neighborhoods, Combin. Probab. Comput., to appear.

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The Turán problem for hypergraphs of fixed size

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