On the KŁR conjecture in random graphs
The KŁR conjecture of Kohayakawa, Łuczak, and Rödl is a statement that allows one to prove that asymptotically almost surely all subgraphs of the random graph , for sufficiently large , satisfy an embedding lemma which complements the sparse regularity lemma of Kohayakawa and Rödl. 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.
Szemerédi’s regularity lemma , which played a crucial role in Szemerédi’s proof of the Erdős-Turán conjecture  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 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 and is said to be -regular if, for every and with and , the density of edges between and satisfies
We will say that a partition of the vertex set of a graph into pieces is an equipartition if, for every , we have the condition that . We say that the partition is -regular if it is an equipartition and, for all but at most pairs , the induced graph between and is -regular. Szemerédi’s regularity lemma can be formally stated as follows.
For every and every positive integer , there exists a positive integer such that every graph with at least vertices admits an -regular partition of its vertex set into 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 a graph contains, in terms of the densities arising in an -regular partition. The so-called regularity method usually works as follows. First, one applies the regularity lemma to a graph . Next, one defines an auxiliary graph whose vertices are the parts of the regular partition of 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 and is the density .) If one can then find a copy of a particular subgraph in , the counting lemma allows one to find many copies of in . If does not contain a copy of , this information can often be used to deduce some further structural properties of the graph and, thereby, the original graph . 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 Erdős-Stone theorem , its stability version due to Erdős and Simonovits , and the graph removal lemma [4, 24, 29, 76].
For sparse graphs – that is, graphs with vertices and edges – the regularity lemma stated in Theorem 1.1 is vacuous, since every equipartition into a bounded number of parts is -regular for sufficiently large. It was observed independently by Kohayakawa  and Rödl 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 of density and a subgraph of of density at least (or relative density at least in ), where usually tends to 0, while is usually independent of the number of vertices.
To make this precise, we say that a bipartite graph between sets and is -regular if, for every and with and , the density of edges between and satisifies
That is, we alter the definition of regularity so that it is relative to a particular density . This density is usually comparable to the total density between and . A partition of the vertex set of a graph into pieces is then said to be -regular if it is an equipartition and, for all but at most pairs , the induced graph between and is -regular.
The class of graphs to which the Kohayakawa-Rödl regularity lemma applies are the so-called upper-uniform graphs . Suppose that , , and are given. We will say that a graph is -upper-uniform if for all disjoint subsets and with , the density of edges between and satisfies . This condition is satisfied for many natural classes of graphs, including all subgraphs of random and pseudorandom graphs of density . The regularity lemma of Kohayakawa and Rödl is the following.
For every and every positive integer , there exist and a positive integer such that for every , every graph with at least vertices that is -upper-uniform admits an -regular partition of its vertex set into 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 -upper-uniformity assumption is altered [3, 63] or dropped completely .
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 and replace its vertices by large independent sets and its edges by -regular bipartite graphs with non-negligible density, then this blown-up graph will contain roughly the expected number of copies of . Here is a precise statement to that effect.
For every graph with vertex set and every , there exists and an integer such that the following statement holds. Let and let be a graph whose vertex set is a disjoint union of sets of size . Assume that for each , the bipartite subgraph of induced between and is -regular and has density . Then the number of -tuples such that whenever is .
In particular, if the density is large for every , then contains many copies of .
Let us define a canonical copy of in to be a -tuple as in the lemma above: that is, a -tuple such that for every and for every . Let us also write for the number of canonical copies of in . (Of course, the definitions of “canonical copy” and depend not just on but also on the partition of into , 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 , which can tend to zero with . We would replace the densities by and we would like to show that is approximately . In order to obtain this stronger conclusion (stronger because the error estimate has been multiplied by ), we need a stronger assumption, and the natural assumption, given the statement of Theorem 1.2 (which is itself natural), is to replace -regularity by -regularity.
Of course, we cannot expect such a result if is too small. Consider the random graph obtained from by replacing each vertex of by an independent set of size and each edge of by a random bipartite graph with edges. With high probability, will be about . Hence, if , then one can remove all copies of from 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 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 will be -regular for some , but that will contain no canonical copies of .
Therefore, a sparse analogue of Lemma 1.3 cannot hold if for some small positive constant . Note that one can replace in the above argument by an arbitrary subgraph , since removing all copies of from a graph also results in a -free subgraph. This observation naturally leads to the notion of -density of a graph , defined by
(We take .) 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 for some absolute constant .
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 of size for some that is much smaller than , and then one can replace each vertex of this small graph by an independent set of vertices to make a graph with vertices in each . Roughly speaking, the counterexample above survives the blowing-up process, and the result is that the hoped-for sparse counting lemma is false whenever . (For more details, see [34, 52].)
However, these “block” counterexamples have a special structure, so, for , it looks plausible that graphs for which the sparse counting lemma fails should be very rare. This intuition was formalized by Kohayakawa, Łuczak, and Rödl , who made a conjecture that is usually known as the KŁR conjecture. Before we state it formally, let us introduce some notation.
As above, let be a graph with vertex set . We denote by the collection of all graphs obtained in the following way. The vertex set of is a disjoint union of sets of size . For each edge , we add to an -regular bipartite graph with edges between the pair . These are the only edges of . Let us also write for the set of all that do not contain a canonical copy of .
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 ). However, as we shall see later, it is sufficient to consider just this case. In fact, the KŁR conjecture is more specific still, since it takes all the densities to be equal to . Again, it turns out that from this case one can deduce the other cases that are needed.
Let be a fixed graph and let . Then there exist and a positive integer such that
for every and every .
Note that is the number of graphs with vertex set with edges between each pair when and no edges otherwise. Thus, we can interpret the conjecture as follows: the probability that a random such graph belongs to the bad set is at most .
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, -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 be a random graph with vertices and edge probability and let and . Then the expected number of subgraphs of of the form is at most
Therefore, choosing to be sufficiently small in terms of , and , the probability that contains a graph in is very small. By summing over the possible values of and , we may rule out such bad subgraphs for all and with and .
This does not give us a counting lemma for -regular subgraphs of , but it does at least tell us that every -regular subgraph of with sufficiently dense pairs in the right places contains a canonical copy of . In other words, it gives us an embedding lemma, which makes it suitable for several applications to embedding results. For example, as noted in , 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 Turán’s theorem for random graphs, which was eventually proved in a different way by Conlon and Gowers  (for strictly balanced graphs, i.e., those for which for every proper subgraph of ) and, independently, Schacht  (see also [10, 78]). We remark that this theorem was the original motivation behind Conjecture 1.4 – see Section 6 of . Following , let us say that a graph is -Turán if every subgraph of with at least
edges contains a copy of . Here is the chromatic number of .
For every and every graph , there exist positive constants and such that
The KŁR 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 that do not contain a cycle. In this case, the class will be empty. The cases , , and were resolved in , , and , respectively. In the case when is a cycle, the conjecture was proved in [11, 30] (see also  for a slightly weaker version). Very recently, it was proved for all balanced graphs, that is, those graphs for which , by Balogh, Morris, and Samotij  and by Saxton and Thomason  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 , including Ramsey’s theorem  and the Erdős-Simonovits stability theorem . However, there are situations where an embedding result is not enough: rather than just a single copy of , 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 KŁR 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 KŁR conjecture but do follow from our counting result.
Our main theorem is the following.
For every graph and every , there exist with the following property. For every , there is a such that if , then a.a.s. the following holds in :
For every , , and every subgraph of in ,
Moreover, if is strictly balanced, that is, if for every proper subgraph of , then
Note that strictly speaking the statements above depend not just on the graph but on the partition that causes to belong to . Roughly speaking, (i) tells us that if contains “many” edges in the right places, then there are “many” copies of , while (ii) tells us that the number of copies of 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.
The proof of part (i) employs the ideas of Schacht , as modified by Samotij , and as a result part (i) holds with probability at least for some depending on , and . Part (ii) is proved using the results of Conlon and Gowers  and hence hold with probability at least for any fixed , provided that and are sufficiently large. Since part (ii) gives an upper bound as well as a lower bound, standard results on upper tail estimates imply that the result cannot hold with the same exponential probability as part (i) (see, for example, ).
We note that weaker versions of Theorem 1.6, applicable for larger values of , may be found in some earlier papers on the KŁR conjecture and Turán’s theorem [31, 54] and more recent work on sparse regularity in pseudorandom graphs . We also believe that a variant of part (i) of Theorem 1.6 may be derivable from the work of Saxton and Thomason , though they have not stated it in these terms.
1.1 Known applications
It is not hard to show that Theorem 1.6, like Conjecture 1.4, implies the best possible sparse random analogues of many classical theorems in extremal graph theory. In particular, it implies Theorem 1.5 above. It also implies the following sparse random version of the Erdős-Simonovits stability theorem, which was first proved by Conlon and Gowers  for all strictly balanced graphs and later extended to general by Samotij , who adapted Schacht’s method for this purpose.
For every graph and every , there exist such that if , then a.a.s. every -free subgraph with may be made -partite by removing at most edges.
Another easy consequence of Theorem 1.6 is the -statement of the following sparse Ramsey theorem, originally proved by Rödl and Ruciński  in 1995. We remark here that the -statement of Theorem 1.8 below also extends to general hypergraphs [17, 28]. Following , we say that a graph is -Ramsey if every -colouring of the edges of yields a monochromatic copy of .
For every graph that is not a star forest or a path of length and for every positive integer , there exist constants such that
1.2 A sparse removal lemma for graphs
The triangle removal lemma of Ruzsa and Szemerédi  states that for every there exists an such that if is any graph on vertices that contains at most triangles, then may be made triangle-free by removing at most edges. Despite its innocent appearance, this result has several striking consequences. Most notably, it easily implies Roth’s theorem  on -term arithmetic progressions in dense subsets of the integers. For general graphs , a similar statement holds [4, 24, 29] (see also [14, 27]): if an -vertex graph contains copies of , then it may be made -free by removing edges. This result is known as the graph removal lemma. A sparse random version of the graph removal lemma was conjectured by Łuczak in  and proved, for strictly balanced , by Conlon and Gowers . Here, we apply our main result, Theorem 1.6, to extend this result to all graphs .
For every and every graph , there exist positive constants and such that if , then the following holds a.a.s. in . Every subgraph of which contains at most copies of may be made -free by removing at most edges.
Note that if for a sufficiently small positive constant (depending on and ), then has a subgraph , the expected number of copies of which is at most , so we can remove all copies of by deleting an edge from each copy of . Thus, it is natural to conjecture, as Łuczak did, that Theorem 1.9 actually holds for all values of . For balanced graphs, we may close the gap by taking to be sufficiently small in terms of , , and . For and , the number of copies of is at most . Deleting an edge from each copy of yields the result.
1.3 The clique density theorem
For any , let be the minimum number of copies of which are contained in any graph on vertices with density at least . We then take
Complete -partite graphs demonstrate that for . On the other hand, a robust version of Turán’s theorem known as supersaturation  tells us that for all .
There has been much work [12, 22, 23, 36, 62] on determining above the natural threshold , but it is only in recent years that the exact dependency of on has been found for any . For and the interval , this was accomplished by Fisher [26, 35], while for a general the value of was determined by Razborov [68, 69] using flag algebras. Employing different methods, Nikiforov  reproved the case and also solved the case . Finally, Reiher  recently resolved the general case.
We will show that an analogous theorem holds within the random graph . We note that this theorem, Theorem 1.10 below, also follows as a direct application of the results of . To state the result, we define, for a subgraph of , the relative density of in to be .
For any and any , there exists a constant such that if , then the following holds a.a.s. in . Any subgraph of will contain at least copies of , where is the relative density of in .
We remark that, as , the assumption on in the above theorem is best possible up to the value of the constant . Using part (ii) of Theorem 1.6, we may show that a similar theorem also holds for any strictly balanced graph . That is, if is the natural analogue of defined for , then for any , there exists a such that if , the random graph will a.a.s. be such that any subgraph of contains at least copies of , where again is the relative density of in . However, the function is only well understood for cliques and certain classes of bipartite graph (see, for example, [15, 61]).
1.4 The Hajnal-Szemerédi theorem
Let be a fixed graph on vertices. An arbitrary collection of vertex-disjoint copies of in some larger graph is called an -packing. A perfect -packing (or -factor) is an -packing that covers all vertices of the host graph. It has long been known for certain graphs that if the minimum degree of an -vertex graph is sufficiently large and is divisible by , then contains an -factor. For example, Dirac’s theorem  implies that if is a path of length , is divisible by , and , then contains an -factor. Corrádi and Hajnal  proved that implies the existence of a -factor in . A milestone in this area of research, the famous theorem of Hajnal and Szemerédi , states that the condition is sufficient to guarantee a perfect -packing in for an arbitrary (see  for a short proof of this theorem).
For any , every -vertex graph with contains a -factor, provided that is divisible by .
This theorem has also been generalized to arbitrary [7, 56, 59]. In particular, a result of Komlós  shows that a parameter known as the critical chromatic number governs the existence of almost perfect packings (i.e., packings covering all but a -fraction of the vertices of the host graph) in graphs of large minimum degree.
We will prove the following approximate version of the Hajnal-Szemerédi Theorem in the random graph .
For any and any , there exists a constant such that if , then the following holds a.a.s. in . Any subgraph of with contains a -packing that covers all but at most vertices.
We remark that the assumption on in the above theorem is best possible up to the value of the constant . Using the result of Komlós , we may show that a similar theorem also holds for any graph . That is, for any and any , there exists a such that if , then a.a.s. every subgraph of satisfying , where is the critical chromatic number of , contains an -packing covering all but at most vertices.
Finally, we remark that the problem of finding perfect packings in subgraphs of random graphs seems to be much more difficult. On the positive side, a best possible sparse random analogue of Dirac’s theorem was recently proved by Lee and Sudakov . On the negative side, it was observed by Huang, Lee, and Sudakov  that for every , there are such that if every vertex of is contained in a triangle and , then a.a.s. contains a spanning subgraph with such that at least vertices of are not contained in a copy of . Therefore, the presence of the set of uncovered vertices in the statement of Theorem 1.12 is indispensable. For further discussion and related results, we refer the reader to [9, 43].
1.5 The Andrásfai-Erdős-Sós theorem
A result of Zarankiewicz  states that if a graph on vertices has minimum degree at least then it contains a copy of . This result follows immediately from Turán’s theorem but is interesting because it has a surprisingly robust stability version, due to Andrásfai, Erdős, and Sós . This theorem states that any -free graph on vertices with minimum degree at least must be -partite.
This result was extended to general graphs by Alon and Sudakov  (see also ), who showed that for any graph and any , every -free graph on vertices with minimum degree at least may be made -partite by deleting edges. We prove a random analogue of this result, as follows.
For every graph and every , there exists such that if , then a.a.s. every -free subgraph with may be made -partite by removing from it at most edges.
More generally, for any and any , we may define
The Andrásfai-Erdős-Sós theorem determines the value of , but there are also some results known for other values of . For example, it is known [13, 41, 45] that and . Our methods easily allow us to transfer any such results about to the random setting. These are approximate results, proving that any -free graph with a certain minimum degree is close to a graph with bounded chromatic number. As noted in , one cannot hope to achieve a more exact result saying that the graph itself has bounded chromatic number.
2 KŁR conjecture via multiple exposure
In this section, we prove part (i) of Theorem 1.6. The proof follows the ideas of [79, 77]. The main idea in these proofs is to expose the random graph in multiple rounds. In this context, this can be traced back to the work of Rödl and Ruciński in .
We are going to prove a somewhat stronger statement, Theorem 2.1 below, which easily implies part (i) of Theorem 1.6. A bipartite graph between sets and is -lower-regular if, for every and with and , the density of edges between and satisfies . Given a graph on the vertex set , we denote by the collection of all graphs on the vertex set , where are pairwise disjoint sets of size each, whose edge set consists of different -lower-regular bipartite graphs, one graph between and for each . For an arbitrary graph and , we will denote by the random subgraph of , where each edge of is included with probability , independently of all other edges. Finally, given and two graphs , we denote by the set of all canonical copies of in such that the edges of in these copies are in .
Let be an arbitrary graph. For every and every , there exist and an integer such that if and , then the following holds. For every , with probability at least , the random graph has the following property: Every subgraph of in satisfies
Proof of part (i) of Theorem 1.6.
Let be an arbitrary -vertex graph and let . We may assume that contains a vertex of degree at least two (and hence ) since otherwise the assertion of the theorem is trivial. Let
Moreover, let and let . Next, fix an arbitrary positive constant and let . Assume that and that is sufficiently large. Finally, let and let . We estimate the probability of the event (we denote it by ) that contains a subgraph with .
Let be pairwise disjoint subsets of , each of size , and let denote the event that contains a subgraph as above with sets playing the role of from the definition of . By Chernoff’s inequality (see below),
On the other hand, if , then by Theorem 2.1 with , and , the complete blow-up of obtained by replacing the vertices of by sets and the edges by complete bipartite graphs, the probability that contains a subgraph satisfying
is at most . To see this, note that has the same distribution as and that our assumptions imply that . It follows that
since . ∎
Note that we used Chernoff’s bound in the following standard form (see, for example, [5, Appendix A]).
Lemma 2.2 (Chernoff’s inequality).
Let be a positive integer, , and . For every positive ,
In particular, if we apply the upper tail estimate with , we see that
as required in the proof above with .
In the proof of Theorem 2.1, we will also need the following approximate concentration result for random subgraphs of , the complete blow-up of obtained by replacing the vertices of by disjoint sets of size each and the edges of by complete bipartite graphs. Lemma 2.3 below is [79, Proposition 3.6] with being the -uniform hypergraph on the vertex set whose edges are the canonical copies of in . The proof of the fact that this hypergraph is -bounded, see [79, Definition 3.2], is implicit in the proof of the -statement of [79, Theorem 2.7]. The definition of is given in Section 2.2.2.
Let be an arbitrary graph with . There exists a such that for every proper and every , there exist and an integer such that for every , if , then with probability at least , for every there exists a subgraph with satisfying
Let be an arbitrary graph and let be a positive constant. In the remainder of this section, we prove Theorem 2.1 by induction on the number of edges in .
2.1 Induction base ()
Let be the empty subgraph of and note that, regardless of , we have . The base of the induction follows immediately from the following one-sided version of the counting lemma, Lemma 1.3, if we let , , , , and .
For every graph and every , there exist such that for every and every ,
In fact, by choosing sufficiently small, one can show that .
2.2 Induction step
Let be an arbitrary subgraph of with edges. Let , , , , and be the constants whose existence is asserted by the inductive assumption with replaced by and replaced by , i.e., let
We also let , , and . Furthermore, let
Throughout the proof, we will assume that , where is sufficiently large. In particular, we will assume that is larger than the values of which come from applying Theorem 2.1 and Lemma 2.3 in context. Finally, assume that satisfies and fix some .
2.2.1 Multiple exposure trick
Let denote the event that the random graph possesses the postulated property:
|:||Every subgraph such that satisfies .|
Following Samotij , we will consider a richer probability space that is in a natural correspondence with the space of all subgraphs of equipped with the obvious probability measure , i.e., the distribution of the random graph . To this end, let be the unique numbers that satisfy
and observe that
The richer probability space will be the space equipped with the product measure that is the distribution of the sequence of independent random variables, where for each , the variable is a -random subgraph of . Crucially, observe that due to our choice of , see (4), the natural mapping
is measure preserving, i.e., for every ,
In other words, the variables and have the same distribution. Finally, let and consider the following event in the space :