Stability and Erdős–Stone type results for -free graphs with a fixed number of edges
A fundamental problem of extremal graph theory is to ask, “What is the maximum number of edges in an -free graph on vertices?” Recently Alon and Shikhelman proposed a more general, subgraph counting, version of this question. They considered the question of determining the maximum number of copies of a fixed graph in an -free graph on vertices.
In this more general context, where we are no longer counting edges, it is also natural to ask what is the maximum number of copies of in an -free graph with edges and no restriction on the number of vertices. Frohmader, in a different context, determined the answer when and are both complete graphs. We prove results for this problem analogous to the Erdős–Stone theorem, the Erdős–Simonovits theorem, and the stability theorem of Erdős–Simonovits.
1.1 Extremal numbers and generalizations of extremal numbers
The fundamental problem of extremal graph theory is to compute the extremal number,
Recently Alon and Shikhelman  proposed a more general version of this problem. Rather than counting edges, they considered the problem of determining the maximum number of copies of a fixed graph in an -free graph on vertices. Letting be the number of copies of in , we define
Turán’s theorem  states that , the number of edges in the Turán graph , the complete -partite graph on vertices with parts as equal in size as possible. Moreover the Turán graph is the unique extremal graph. The following result was proved by Zykov  (and has since been rediscovered many times).
For all , the maximum in the definition of is uniquely achieved by the Turán graph . In other words,
Now that we are counting copies of , rather than edges, it makes sense to shift away from our resource being a limited number of vertices we are allowed, and consider similar problems for the class of graphs with edges. We make a third parallel definition.
It is important to note that this definition does not place any restriction on the number of vertices of .
1.2 Previous results about
Some results about are known (though not using that terminology). One can even think of the Kruskal–Katona theorem [11, 12] as proving a result in this direction. We start with a little background about that theorem.
Given , , let denote the family of -sets of . The colexicographic or colex order on is defined as follows: for all , , if and only if . For a family we define the shadow of on level to be the set
The Kruskal–Katona theorem gives a bound for the minimum size of as a function of the size of .
If and is the colex initial segment of of size then for any we have
One should also note is itself an initial segment in the colex order on .
It is an immediate corollary of Theorem 2 that for every and , the maximum number of copies of in a graph with edges is achieved by the graph with vertex set whose edge set consists of the first elements of in colex order. We call this graph the colex graph with edges, and denote it . This is a slight abuse of notation, since we have not specified , but in our problems we only care that we have enough vertices, not how many there actually are.
Frohmader  determined the value of for all . His results were phrased in terms of simplicial complexes, so let us take a moment to recall the relevant definitions.
Let be a simplicial complex. If is a face of , then the dimension of is . The dimension of is . Let and, for each , , let denote the number of -dimensional faces in . Recall that the -vector of is the -tuple . We say that a complex is -colorable if there is a partion of its vertex set into parts such that each set in meets each part in at most one element.
A simplicial complex is called a flag complex if every minimal non-face of has two elements. This is equivalent to the notion of a “clique complex”: the clique complex of a graph is the simplicial complex whose vertex set is and whose faces are the cliques of . It is easy to see that a flag complex is -colorable if and only if it is the clique complex of an -colorable graph. We say that a complex is balanced if and is -colorable.
Kalai (unpublished; see [15, p. 100]) and Eckhoff  conjectured that if is a flag complex, then there exists a balanced complex with the same -vector as . Frohmader  proved the Kalai–Eckhoff conjecture. This is in fact sufficient to determine . For completeness we include a proof of this deduction below.
We will need to quote the “colored” version of the Kruskal–Katona theorem, proved by Frankl, Füredi, and Kalai . The role played by the colex order in the Kruskal–Katona theorem is played here by the -partite colex order. This is colex order restricted to subsets of such that for all we have or .
Given and , the colex Turán graph is the graph on vertex set whose edge set consists of the first edges in -partite colex order. (See Figure 1.) Note that if , then the unique non-trivial component of is isomorphic to .
Theorem 3 ().
If is -colorable and is the initial segment of in the -partite colex order of size , then for any we have
One should also note is itself an initial segment in the -partite colex order on .
In the next corollary and throughout the rest of the paper we write for . Also, given and , let and denote the number of copies of in that contain and the number of copies of in that contain , respectively.
If is an -partite graph with edges then
This is an immediate consequence of Theorem 3 and the definition of . ∎
From these results we can prove that achieves the maximum in the definition of .
Theorem 5 (Frohmader ).
For all we have
Consider a -free graph having edges. Its clique complex is a flag complex, so, since the Kalai–Eckhoff conjecture is true, there is a balanced complex having the same -vector as . Since we know that is -colorable. By Corollary 4 we have . ∎
1.3 Structural supersaturation and stability
Turán’s theorem has inspired a great deal of research on the size and structure of extremal and near-extremal -free graphs. We mention several important theorems in this area in order to motivate our results.
Given a graph and a positive integer , we let denote the -fold blowup of (in which every vertex of is replaced by an independent set of size and every edge by a copy of ). Erdős and Stone  showed that a graph with vertices and edges not only contains , but contains a sizable blow-up of .
For all , , and , there exists such that if and is a graph with vertices such that
then contains .
Let be a graph. We have
For all and all , there exist and such that if and is a -free graph with vertices such that
then can be made -partite by deleting at most edges.
In Section 2 we prove analogues of the Erdős–Stone theorem (Theorem 6), the Erdős–Simonovits theorem (Theorem 7), and the Erdős–Simonovits stability theorem (Theorem 8) in the context of Frohmader’s theorem, Theorem 5. To be precise we prove the following results.
For all , all , and all , there exists such that if and is a graph with edges such that
then contains .
Let , let be a graph, and let . We have
For all and all , there exist and such that if and is a -free graph with edges such that
then can be made -partite by deleting at most edges.
For all , every graph with chromatic number , and all , there exist and such that if and is an -free graph with edges such that
then can be made -partite by deleting at most edges.
Our results establish a number of parallels between and , with the colex Turán graph playing the role in results about that the Turán graph plays in results about . However, this correspondence is not perfect.
Let be a graph with a critical edge. Simonovits  used the stability method to determine (and the extremal graph) for all sufficiently large.
Let be a graph with and suppose that contains an edge such that . For all sufficiently large, and is the unique extremal graph.
In contrast, if is as in the statement of Theorem 13 and , there exist infinitely many values of such that is not an extremal graph for .
Given , let be the least integer such that . Let be the graph consisting of and a vertex that is joined to vertices of , distributed as evenly as possible among the classes of . Observe that if , then is -free but not -partite. Moreover,
Finally there are a number of very natural analogues of results concerning that are open for . In Section 3 we briefly discuss some of these open problems.
2.1 Preliminaries and notation
Let be a graph and let . Recall that for and , and denote the number of copies of in that contain and the number of copies of in that contain , respectively. The minimum values of these quantities are denoted
In the extremal -free graph the average degree is a multiple of and the number of copies of is a multiple of . We define those constant multiples here. Given and , let
The following simple proposition collects some computations about .
If and , then and so
In particular in this case is -regular. Furthermore we have
We also record some properties of the constants and defined above.
For all and ,
We will need a consequence of the Kruskal–Katona theorem noted by Lovász [13, Exercise 13.31].
Let and let . If is a graph with edges, then .
Let . If is a graph with edges, then
We will also need the following result of Erdős, Frankl, and Rödl .
Let . For all and every graph with chromatic number , there exists such that if is an -free graph of order , then can be made -free by removing at most edges.
2.2 Proof of Theorem 9
Proof of Theorem 9.
First, we show that if is sufficiently large, then contains a subgraph that has both positive edge density and many copies of relative to .
Let be such that
Let be sufficiently large. If , we do nothing. Otherwise, we let and, for each , if contains an edge with , we set .
Suppose that we delete such edges and let denote the resulting subgraph. We have
Thus, using (5) twice, we have
which contradicts Corollary 17.
So, has a subgraph with edges and vertices such that
We claim that
Indeed, given , suppose that . If is sufficiently small, then we have
The claimed inequality (6) follows by induction on and our assumption on .
Observe that if and is an endpoint of , then . It follows that
Suppose that does not contain a copy of . By the trivial bound , we may let be as large as we wish by taking to be sufficiently large. So, by (7) and Theorem 18, if is sufficiently large, then we can delete all copies of in by removing at most edges. This means that we remove at most copies of . Let denote the resulting graph and let . By (6), Proposition 14, and Theorem 5, we have
a contradiction. ∎
2.3 Proof of Theorem 11
Proofs of stability results in extremal graph theory often begin by showing that a global density assumption on a graph implies a minimum degree condition. This is frequently accomplished by iteratively deleting vertices of degree at most (where is an appropriate constant) and showing that the density of and the forbidden subgraph condition mean that only a small fraction of the vertices could have been deleted in this way.
However, in our case, if we delete vertices whose degree is too small as a function of the number of vertices of , then there is no reason to expect that the process will terminate quickly, for the simple reason that we do not know how many vertices has. In particular, we may end up deleting far more than edges. Instead, letting denote the set of the vertices of whose degree is too small as a function of the number of edges of , we will show that the vertices of span only a small fraction of the edges of . We will then be able to show that has high minimum degree as a function of the number of vertices.
Given and , there exist and with the following property. If and is a -free graph with edges such that
then has a subgraph with vertices and edges such that
Let be sufficiently small and let
Let . For each , if contains a vertex with , set . Suppose that we delete edges in this way and that we delete edges incident to vertices. (To ensure that we delete exactly edges, if necessary we do not delete the final vertex , but instead delete the appropriate number of edges incident to it.) Let denote the resulting graph. We have
Because is -free, for each , is -free. Hence, the number of copies of in that contain is at most . By Theorem 1, for all ,
So, we have
By assumption, for all , . Moreover, by the definition of ,
Combining this bound with (9) gives
which contradicts (2.3).
Proof of Theorem 11.
Given , let
Let be the graph obtained by inputting , , and into Lemma 19 and let .
By Lemma 19,
So, if is sufficiently small, Turán’s theorem implies that contains a copy of with vertex set . Because is -free, every vertex of has at most neighbors in . Let and let . By definition,
On the other hand, by Lemma 19,
It follows that
For , …, , let . It is easy to see that the partition and that each is an independent set. So, if we delete all of the vertices of from , the resulting graph is -partite.
It remains to show that deleting the vertices of from removes only a small number of edges. By Lemma 19, we have , which means that
It follows from (12) that we have deleted at most edges of . This completes the proof. ∎
Proof of Corollary 12.