arXiv:1305.2516v1 [math.CO] 11 May 2013
On the K LR conjecture in random graphs
D. Conlon
∗
W. T. Gowers
†
W. Samotij
‡
M. Schacht
§
Abstract
The K LR conjecture of Kohayakawa, Luczak, and R¨odl is a statement that allows one to
prove that asymptotically almost surely all subgraphs of the random graph G
n,p
, for sufficiently
large p := p(n), satisfy an embedding lemma which complements the sparse regularity lemma of
Kohayakawa and R¨odl. We prove a variant of this conjecture which is sufficient for most known
applications to random graphs. In particular, our result implies a number of recent probabilistic
versions, due to Conlon, Gowers, and Schacht, of classical extremal combinatorial theorems. We
also discuss several further applications.
1 Introduction
Szemer´edi’s regularity lemma [83], which played a crucial role in Szemer´edi’s proof of the Erd˝os-
Tur´an conjecture [82] on long arithmetic progressions in dense subsets of the integers, is one of the
most important tools in extremal graph theory (see [57, 58, 73]). Roughly speaking, it says that
the vertex set of every graph G may be divided into a bounded number of parts in such a way that
most of the induced bipartite graphs between different parts are pseudorandom.
More precisely, a bipartite graph between sets U and V is said to be ǫ-regular if, for every
U
′
⊆ U and V
′
⊆ V with |U
′
| ≥ ǫ|U| and |V
′
| ≥ ǫ|V |, the density d(U
′
, V
′
) of edges between U
′
and V
′
satisfies
|d(U
′
, V
′
) − d(U, V )| ≤ ǫ.
We will say that a partition of the vertex set of a graph into t pieces V
1
, . . . , V
t
is an equipartition
if, for every 1 ≤ i, j ≤ t, we have the condition that ||V
i
| − |V
j
|| ≤ 1. We say that the partition
is ǫ-regular if it is an equipartition and, for all but at most ǫt
2
pairs (V
i
, V
j
), the induced graph
between V
i
and V
j
is ǫ-regular. Szemer´edi’s regularity lemma can be formally stated as follows.
Theorem 1.1. For every ǫ > 0 and every positive integer t
0
, there exists a positive integer T such
that every graph G with at least t
0
vertices admits an ǫ-regular partition V
1
, . . . , V
t
of its vertex set
into t
0
≤ t ≤ T pieces.
Often the strength of the regularity lemma lies in the fact that it may be combined with a
counting or embedding lemma that tells us approximately how many copies of a particular subgraph
∗
Mathematical Institute, Oxford OX1 3LB, United Kingdom. E-mail: david.conlon@maths.ox.ac.uk. Research
supported by a Royal Society University Research Fellowship.
†
Department of Pure Mathematics an d Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB, UK.
E-mail: w.t.gowers@dpmms.cam.ac.uk. Research supported by a Royal Society 2010 Anniversary Research Profes-
sorship.
‡
School of Mathematical Sciences, Tel Aviv University, Tel A viv, I srael; and Trinity College, Cambridge CB2
1TQ, UK. E-mail: samotij@post.tau.ac.il. Research sup ported in part by a Trinity College JRF.
§
Fachbereich Mathematik, Universit¨at Hamburg, Bundesstraße 55, D-20146 Hamburg, Germany. E-mail:
schacht@math.uni-hamburg.de. Research supported by the Heisenberg p rogramme of the DFG.
1
a graph contains, in terms of the densities d(V
i
, V
j
) arising in an ǫ-regular partition. The so-called
regularity method usually works as follows. First, one applies the regularity lemma to a graph G.
Next, one defines an auxiliary graph R whose vertices are the parts of the regular partition of G
one obtains, and whose edges correspond to regular pairs with non-negligible density. (For some
applications we may instead take a weighted graph, where the weight of the edge between a regular
pair V
i
and V
j
is the density d(V
i
, V
j
).) If one can then find a copy of a particular subgraph H
in R, the counting lemma allows one to find many copies of H in G. If R does not contain a
copy of H, this information can often be used to deduce some further structural properties of the
graph R and, thereby, the original graph G. Applied in this manner, the regularity and counting
lemmas allow one to prove a number of well-known theorems in extremal graph theory, including
the Erd˝os-Stone theorem [21], its stability version due to Erd˝os and Simonovits [81], and the graph
removal lemma [4, 24, 29, 76].
For sparse graphs – that is, graphs with n vertices and o(n
2
) edges – the regularity lemma stated
in Theorem 1.1 is vacuous, since every equipartition into a bounded number of parts is ǫ-regular for
n sufficiently large. It was observed independently by Kohayakawa [47] and R¨odl that the regularity
lemma can nevertheless be generalized to an appropriate class of graphs with density tending to
zero. Their result applies to a natural class of sparse graphs that is wide enough for the lemma
to have several interesting applications. In particular, it applies to relatively dense subgraphs of
random graphs – that is, one takes a random graph G
n,p
of density p and a subgraph G of G
n,p
of
density at least δp (or relative density at least δ in G
n,p
), where p usually tends to 0, while δ > 0
is usually independent of the number of vertices.
To make this precise, we say that a bipartite graph between sets U and V is (ǫ, p)-regular if, for
every U
′
⊆ U and V
′
⊆ V with |U
′
| ≥ ǫ|U| and |V
′
| ≥ ǫ|V |, the density d(U
′
, V
′
) of edges between
U
′
and V
′
satisifies
|d(U
′
, V
′
) − d(U, V )| ≤ ǫp.
That is, we alter the definition of regularity so that it is relative to a particular density p. This
density is usually comparable to the total density between U and V . A partition of the vertex set
of a graph into t pieces V
1
, . . . , V
t
is then said to be (ǫ, p)-regular if it is an equipartition and, for
all but at most ǫt
2
pairs (V
i
, V
j
), the induced graph between V
i
and V
j
is (ǫ, p)-regular.
The class of graphs to which the Kohayakawa-R¨odl regularity lemma applies are the so-called
upper-uniform graphs [53]. Suppose that 0 < η ≤ 1, D > 1, and 0 < p ≤ 1 are given. We will
say that a graph G is (η, p, D)-upper-u ni form if for all disjoint subsets U
1
and U
2
with |U
1
|, |U
2
| ≥
η|V (G)|, the density of edges between U
1
and U
2
satisfies d(U
1
, U
2
) ≤ Dp. This condition is satisfied
for many natural classes of graphs, including all subgraphs of random and pseudorandom graphs
of density p. The regularity lemma of Kohayakawa and R¨odl is the following.
Theorem 1.2. For every ǫ, D > 0 and every positive integer t
0
, there exist η > 0 and a positive
integer T such that for every p ∈ [0, 1], every graph G with at least t
0
vertices that i s (η, p, D)-
upper-uniform admits an (ǫ, p)-regular partition V
1
, . . . , V
t
of its vertex set into t
0
≤ t ≤ T pieces.
The proof of this theorem is essentially the same as the proof of the dense regularity lemma,
with the upper uniformity used to ensure that the iteration terminates after a constant number
of steps. We note that different versions of the result have appeared in the literature where the
(η, p, D)-upper-uniformity assumption is altered [3, 63] or dropped completely [80].
As we have already mentioned above, the usefulness of the dense regularity method relies on
the existence of a corresponding counting lemma. Roughly speaking, a counting lemma says that if
we start with an arbitrary graph H and replace its vertices by large independent sets and its edges
by ǫ-regular bipartite graphs with non-negligible density, then this blown-up graph will contain
roughly the expected number of copies of H. Here is a precise statement to that effect.
2
Lemma 1.3. For every graph H with vertex set {1, 2, . . . , k} and every δ > 0, there exists ǫ > 0
and an integer n
0
such that the following statement holds. Let n ≥ n
0
and let G be a graph whose
vertex set is a disjoint union V
1
∪ . . . ∪ V
k
of sets of size n. Assume that for each ij ∈ E(H),
the bipartite subgraph of G induced between V
i
and V
j
is ǫ-regular and has density d
ij
. Then the
number of k-tuples (v
1
, . . . , v
k
) ∈ V
1
× · · · × V
k
such that v
i
v
j
∈ E(G) whenever ij ∈ E(H) is
n
k
(
Q
ij∈E(H)
d
ij
± δ).
In particular, if the density d
ij
is large for every ij ∈ E(H), then G contains many copies of H.
Let us define a canonical copy of H in G to be a k-tuple as in the lemma above: that is, a
k-tuple (v
1
, . . . , v
k
) such that v
i
∈ V
i
for every i ∈ V (H) and v
i
v
j
∈ E(G) for every ij ∈ E(H).
Let us also write G(H) for the number of canonical copies of H in G. (Of course, the definitions
of “canonical copy” and G(H) depend not just on G but also on the partition of G into V
1
, . . . , V
k
,
but we shall suppress this dependence in the notation.)
In order to use Theorem 1.2, one would ideally like a statement similar to Lemma 1.3 but
adapted to a sparse context. For this we would have an additional parameter p, which can tend
to zero with n. We would replace the densities d
ij
by d
ij
p and we would like to show that G(H)
is approximately n
k
p
e(H)
(
Q
ij∈E(H)
d
ij
± δ). In order to obtain this stronger conclusion (stronger
because the error estimate has been multiplied by p
e(H)
), we need a stronger assumption, and the
natural assumption, given the statement of Theorem 1.2 (which is itself natural), is to replace
ǫ-regularity by (ǫ, p)-regularity.
Of course, we cannot expect such a result if p is too small. Consider the random graph G
obtained from H by replacing each vertex of H by an independent set of size n and each edge
of H by a random bipartite graph with pn
2
edges. With high probability, G(H) will be about
p
e(H)
n
v(H)
. Hence, if p
e(H)
n
v(H)
≪ pn
2
, then one can remove all copies of H from G by deleting
a tiny proportion of all edges. (We may additionally delete a further small proportion of edges to
ensure that all bipartite graphs corresponding to the edges of H have the same number of edges.)
It is not hard to see that with high probability the bipartite graphs that make up the resulting
graph G
′
will be (ǫ, q)-regular for some q = (1− o(1))p, but that G
′
will contain no canonical copies
of H.
Therefore, a sparse analogue of Lemma 1.3 cannot hold if p ≤ cn
−
v(H)−2
e(H)−1
for some small positive
constant c. Note that one can replace H in the above argument by an arbitrary subgraph H
′
⊆ H,
since removing all copies of H
′
from a graph also results in a H-free subgraph. This observation
naturally leads to the notion of 2-density m
2
(H) of a graph H, defined by
m
2
(H) = max
e(H
′
) − 1
v(H
′
) − 2
: H
′
⊆ H with v(H
′
) ≥ 3
.
(We take m
2
(K
2
) =
1
2
.) With this notation, what we have just seen is that to have any chance of
an appropriate analogue of Lemma 1.3 holding, we need to assume that p ≥ Cn
−1/m
2
(H)
for some
absolute constant C > 0.
Unfortunately, there is a more fundamental difficulty with finding a sparse counting lemma
to match a sparse regularity lemma. Instead of sparse random graphs with many vertices, one
can consider blow-ups of sparse random graphs with far fewer vertices. That is, one can pick a
counterexample of the kind just described but with the sets V
i
of size r for some r that is much
smaller than n, and then one can replace each vertex of this small graph by an independent set of
n/r vertices to make a graph with n vertices in each V
i
. Roughly speaking, the counterexample
above survives the blowing-up process, and the result is that the hoped-for sparse counting lemma
is false whenever p = o(1). (For more details, see [34, 52].)
3
However, these “block” counterexamples have a special structure, so, for p ≥ Cn
−1/m
2
(H)
, it
looks plausible that graphs for which the sparse counting lemma fails should be very rare. This
intuition was formalized by Kohayakawa, Luczak, and R¨odl [50], who made a conjecture that is
usually known as the K LR conjecture. Before we state it formally, let us introduce some notation.
As above, let H be a graph with vertex set {1, 2, . . . , k}. We denote by G(H, n, m, p, ǫ) the
collection of all graphs G obtained in the following way. The vertex set of G is a disjoint union
V
1
∪ . . . ∪ V
k
of sets of size n. For each edge ij ∈ E(H), we add to G an (ǫ, p)-regular bipartite
graph with m edges between the pair (V
i
, V
j
). These are the only edges of G. Let us also write
G
∗
(H, n, m, p, ǫ) for the set of all G ∈ G(H, n, m, p, ǫ) that do not contain a canonical copy of H.
Since the sparse regularity lemma yields graphs with varying densities between the various
pairs of vertex sets, it may seem surprising that we are restricting attention to graphs where all the
densities are equal (to m/n
2
). However, as we shall see later, it is sufficient to consider just this
case. In fact, the K LR conjecture is more specific still, since it takes all the densities to be equal
to p. Again, it turns out that from this case one can deduce the other cases that are needed.
Conjecture 1.4. Let H be a fixed graph and let β > 0. Then there exist C, ǫ > 0 and a positive
integer n
0
such that
|G
∗
(H, n, m, m/n
2
, ǫ)| ≤ β
m
n
2
m
e(H)
for every n ≥ n
0
and every m ≥ Cn
2−1/m
2
(H)
.
Note that
n
2
m
e(H)
is the number of graphs with vertex set V
1
∪ · · · ∪ V
k
with m edges between
each pair (V
i
, V
j
) when ij ∈ E(H) and no edges otherwise. Thus, we can interpret the conjecture
as follows: the probability that a random such graph belongs to the bad set G
∗
(H, n, m, m/n
2
, ǫ)
is at most β
m
.
The rough idea of the conjecture is that the probability that a graph is bad is so small that a
simple union bound tells us that with high probability a random graph does not contain any bad
graph – which implies that we may use the sparse embedding lemma we need. In other words, (ǫ, p)-
regularity on its own does not suffice, but if you know in addition that your graph is a subgraph of
a sparse random graph, then with high probability it does suffice.
More precisely, let G be a random graph with N vertices and edge probability p and let n = ηN
and m = dpn
2
≥ n. Then the expected number of subgraphs of G of the form G
∗
(H, n, m, p, ǫ) is
at most
p
me(H)
β
m
n
2
m
e(H)
N
n
v(H)
≤ p
me(H)
β
m
e
dp
me(H)
e
η
nv(H)
≤ β
m
e
d
me(H)
e
η
mv(H)
.
Therefore, choosing β to be sufficiently small in terms of d, η, and H, the probability that G contains
a graph in G
∗
(H, n, m, p, ǫ) is very small. By summing over the possible values of n and m, we may
rule out such bad subgraphs for all n and m with n ≥ ηN and m ≥ dpn
2
.
This does not give us a counting lemma for (ǫ, p)-regular subgraphs of G, but it does at least tell
us that every (ǫ, p)-regular subgraph of G with sufficiently dense pairs in the right places contains a
canonical copy of H. In other words, it gives us an embedding lemma, which makes it suitable for
several applications to embedding results. For example, as noted in [50], it is already sufficiently
strong that a straightforward application of the sparse regularity lemma then allows one to derive
the following theorem, referred to as Tur´an’s theorem for random graphs, which was eventually
proved in a different way by Conlon and Gowers [17] (for strictly balanced graphs, i.e., those for
which m
2
(H) > m
2
(H
′
) for every proper subgraph H
′
of H) and, independently, Schacht [79] (see
4
also [10, 78]). We remark that this theorem was the original motivation behind Conjecture 1.4 –
see Section 6 of [50]. Following [17], let us say that a graph G is (H, ǫ)-Tu r´an if every subgraph of
G with at least
1 −
1
χ(H) − 1
+ ǫ
e(G)
edges contains a copy of H. Here χ(H) is the chromatic number of H.
Theorem 1.5. For every ǫ > 0 and every graph H, there exist positive constants c and C such
that
lim
n→∞
P
G
n,p
is (H, ǫ)-Tur´an
=
(
0, if p < cn
−1/m
2
(H)
,
1, if p > Cn
−1/m
2
(H)
.
The K LR conjecture has attracted considerable attention over the past two decades and has
been verified for a number of small graphs. It is straightforward to verify that it holds for all
graphs H that do not contain a cycle. In this case, the class G
∗
(H, n, m, p, ǫ) will be empty. The
cases H = K
3
, K
4
, and K
5
were resolved in [49], [32], and [33], respectively. In the case when H
is a cycle, the conjecture was proved in [11, 30] (see also [48] for a slightly weaker version). Very
recently, it was proved for all balanced graphs, that is, those graphs H for which m
2
(H) =
e(H)−1
v(H)−2
,
by Balogh, Morris, and Samotij [10] and by Saxton and Thomason [78] in full generality.
Besides implying Theorem 1.5, Conjecture 1.4 is also sufficient for transferring many other
classical extremal results about graphs to subgraphs of the random graph G
n,p
, including Ramsey’s
theorem [67] and the Erd˝os-Simonovits stability theorem [81]. However, there are situations where
an embedding result is not enough: rather than just a single copy of H, one needs to know that
there are many copies. That is, one needs something more like a full counting lemma. In this
paper, we shall state and prove such a “counting version” of the K LR conjecture for subgraphs
of random graphs. Later in the paper we shall give examples of classical theorems whose sparse
random versions do not follow from the K LR conjecture but do follow from our counting result.
Our main theorem is the following.
Theorem 1.6. For every graph H and every δ, d > 0, there exist ǫ, ξ > 0 with the following
property. For every η > 0, there is a C > 0 such that if p ≥ CN
−1/m
2
(H)
, then a.a.s. the following
holds i n G
N,p
:
(i) For every n ≥ ηN, m ≥ dpn
2
, and every subgraph G of G
N,p
in G(H, n, m, p, ǫ),
G(H) ≥ ξ
m
n
2
e(H)
n
v(H)
. (1)
(ii) Moreover, if H is strictly balanced, that is, if m
2
(H) > m
2
(H
′
) for every proper subgraph H
′
of H, then
G(H) = (1 ± δ)
m
n
2
e(H)
n
v(H)
. (2)
Note that strictly speaking the statements above depend not just on the graph G but on the
partition V
1
∪ · · · ∪ V
k
that causes G to belong to G(H, n, m, p, ǫ). Roughly speaking, (i) tells us
that if G contains “many” edges in the right places, then there are “many” copies of H, while (ii)
tells us that the number of copies of H is roughly what one would expect for a random graph with
pairs of the same densities. We note that a result similar to (ii) holds for all graphs if one is willing
to allow some extra logarithmic factors. We will say more about this in the concluding remarks.
5