An extension of Tur\'an's Theorem, uniqueness and stability

We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$. For $(r-1)|M|\ge n$, the $(r-1)$-partite Turan graph turns out to be the unique extremal graph. For $(r-1)|M|<n$, there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results.


Introduction
Turán's Theorem [10], whose proof marks the beginning of Extremal Graph Theory, determines the maximum number of edges of n-vertex graph without a copy of the r-clique K r . It turns out that the unique extremal graph for this problem is the Turán graph T r−1 (n), that is, the complete balanced (r − 1)-partite graph on n-vertices. We write t r−1 (n) to denote the number of edges of T r−1 (n).
Turán's Theorem is a primal example of a stable result: The Erdős-Simonovits stability theorem [4,9] asserts that any n-vertex K r -free graph with almost t r−1 (n) edges looks very similar to T r−1 (n). In order to make this more precise we need the following definition. We say that an n-vertex graph G is ε-close to a graph H on the same vertex set if the edit distance 1 between G and H is at most εn 2 , that is, if H can be obtained from G by editing (deleting/inserting) at most εn 2 edges and relabelling the vertices. In this case we also say that G is (εn 2 )-near to H.
In fact, Erdős and Simonovits both proved more general statements, allowing any fixed r-partite graph H in place of K r . Moreover, in more recent years strengthenings have been proved, for example that most vertices of any G as in Theorem 1 are in an induced (r − 1)-partite graph, [7]. There are also further generalisations, such as obtaining the same conclusion as in Theorem 1 while allowing the size of the forbidden subgraph H to depend on v(G), [8].
A main motivation for proving stability results for extremal statements is that they are often useful in applications where the original extremal statement would not suffice. This is for example the case when the Szemerédi Regularity Lemma (see, e.g., the survey [5]) is used. A prominent example of such an application is the enumeration result of Balogh, Bollobás and Simonovits [3] giving a precise count of H-free graphs. It is worth observing that in most applications the 'basic' stability theorem of Erdős and Simonovits, Theorem 1, suffices.
Our goal is to extend Turán's Theorem, by determining the maximum number of edges in an n-vertex graph G such that no copy of K r in G touches a fixed vertex set M ⊆ V (G) of size m. It turns out that for (r−1)m ≥ n the unique extremal graph is T r−1 (n). The case (r−1)m < n is more complicated. In particular, there is a whole family of extremal graphs, which we describe in Section 1.1 below. In both cases we shall denote the (family of) extremal graphs by T r−1 (n, m), and their number of edges by Our two main results are as follows.
Theorem 2. Given r ≥ 3 and m ≤ n, let G be any n-vertex graph and M ⊆ V (G) contain m vertices, such that no copy of K r in G intersects M . Then (a ) e(G) ≤ t r−1 (n, m), and (b ) if e(G) = t r−1 (n, m) then G ∈ T r−1 (n, m).
Theorem 2(b ) states that the graphs T r−1 (n, m) we construct below are the only extremal graphs. The following theorem provides a corresponding stability result.
Theorem 3. Given r ≥ 3 and ε > 0 there exists γ > 0 such that the following holds. Let m ≤ n, let G be any n-vertex graph and M ⊆ V (G) contain m vertices, such that no copy of K r in G intersects M . If e(G) > t r−1 (n, m) − γn 2 , then G is ε-close to a graph from T r−1 (n, m) in which no copies of K r intersect M .
We remark that Theorem 2(a ) is also included in our previous paper [2], but we did not determine the family of extremal graphs there. 2 Hence our main contribution here is to determine the extremal graphs and prove stability. This, however, turns out to be an important tool for [1], where we determine the maximum number of edges in an n-vertex graph without a given number of vertex-disjoint triangles. Note that the statement of Theorem 3 gives a slightly stronger version of stability than the usual one, namely that the set M is not changed in transforming G to a member of T r−1 (n, m). We require this in [1].
We note that the proof of Theorem 2(a ) as given in [2] hints the main arguments involved in our proof of Theorem 2. However, several additional tweaks and tricks are needed, in particular in the case n > (r − 1)m. We give an outline of the proofs of Theorem 2 and 3 in Section 2.1.
The (r−1)m ≥ n case of Theorem 2 shows that the assumption in Turán's Theorem (or in that of Theorem 1) can be substantially weakened from forbidding K r -copies on all possible r-subsets of the vertex set V (G), to just forbidding K r -copies on a particular family S of r-subsets-the family S which contains all r-subsets of V (G) which intersect M . In [2] we investigated such weakenings of the assumption in Turán's theorem also from a probabilistic perspective. In particular, we proved that forbidding K r -copies on a random family of r-sets S ⊆ n r of size only |S| = O(n 3 ) suffices.
1.1. Extremal graphs. The family T r−1 (n, m) is defined as follows. As previously stated, if n ≤ (r − 1)m then T r−1 (n, m) = T r−1 (n) . So assume from now on that n > (r − 1)m. We explicitly describe the construction of the graphs in T r−1 (n, m). We start with the Turán graph T r−1 ((r − 1)m), with colour classes V 1 , . . . , V r−1 , and an arbitrary set M of m vertices in V 1 ∪ · · · ∪ V r−1 . We add r − 1 new vertices v 1 , . . . , v r−1 to this graph with the following property. For each i ∈ [r − 1], the vertex v i is adjacent to all old and new vertices except those in V i (and itself). Finally, we add a set Y of n − (r − 1)m new vertices each of which is adjacent to all old and new vertices except those in M (and itself). In this way we obtain an (n + r − 1)-vertex graph, which we call G r (n, M ). Note that the graph G r (n, M ) depends on the placement of M in V 1 ∪ · · · ∪ V r−1 . We let T r−1 (n, m) be the family of n-vertex graphs which can be obtained from some graph G r (n, M ) by deleting any r − 1 vertices from {v 1 , . . . , v r−1 } ∪ Y (see also Figure 1.1). We call the vertices v 1 , . . . , v r−1 sporadic. 2 Actually, at the time of writing [2] we believed that the family of extremal graphs described there was complete. Only later we discovered further constructions involving 'sporadic vertices' (see below). Figure 1. Examples of graphs from T 2 (n, m) for n > 2m, with no, one, and two sporadic vertices. Grey depicts complete (bipartite) graphs, the set M is hatched.
Observe that there is no copy of K r in G r (n, M ) which uses vertices of M . Furthermore, the vertices {v 1 , . . . , v r−1 } ∪ Y form a clique in G r (n, M ), and each of these vertices has degree n + r − 2 − m. It follows that indeed every graph in T r−1 (n, m) has the same number of edges, and that number is as desired.

Proofs of Theorem 2 and Theorem 3
2.1. Outline of the proofs. We prove Theorem 2 and Theorem 3 together. We refer to the cases n ≤ (r − 1)m and n > (r − 1)m as Cases I and II, respectively. We prove Case I first, and then prove Case II using Case I. In Case I, we sequentially pick maximum vertex disjoint cliques P 1 , . . . , P k of order at least r. Because of their sizes, we know they do not intersect the set M . A counting argument gives an upper bound on the number of edges in G, depending on the sizes of these cliques (see Lemma 4). This upper bound is enough to prove Theorem 2 in Case I. Further, we infer from Lemma 4 that if e(G) > t r−1 (n) − o(n 2 ) then the total order of the cliques P 1 , . . . , P k must be o(n). Therefore, , and the Erdős-Simonovits Stability Theorem, Theorem 1, applies to the graph G − i P i . Thus, the graph T r−1 (n − o(n)) is similar to G − i P i , which in turn is similar to G, as needed.
Let us note that even though Theorem 2 extends Turán's theorem, the counting argument in Lemma 4 actually relies on Turán's result.
The proof strategy for Theorem 2 in Case II comes naturally from the structure of the extremal graphs. The key property to observe is that in these graphs, the neighborhood of the set M induces essentially T r−1 ((r − 1)|M |) (with the exception of the sporadic vertices). As a first step, we apply a Zykov-type symmetrisation to our graph G with no copy of K r intersecting M (Lemma 5) to obtain a graph G ′ . We then perform a further simple transformation to remove any sporadic vertices, obtaining a graph G ′′ with at least as many edges as G. Now we can show that the union of M and its neighbourhood in G ′′ cover at most (r − 1)|M | vertices (Lemma 6). This means that we can apply the bound from Case I on the union of M and its neighbourhood, and trivial bounds on edges in other parts of the graph, to conclude part (a ). To prove (b ) we observe that equality is possible only if the union of M and its neighbourhood in G ′′ is T r−1 ((r − 1)|M |) by Case I of part (b ) and the trivial bounds are sharp, in which case G ′′ is in T r−1 (n, m). This implies that G ′ is also in T r−1 (n, m) (Lemma 7), and finally we conclude that e(G) = e(G ′ ) only if G = G ′ (Lemma 5), as required.
The proof of Case II of Theorem 3 follows the same pattern as the uniqueness result, using Case I of Theorem 3 to show that G ′′ is close in edit distance to a graph in T r−1 (n, m) and then we show that the same is true of G (Lemma 5).
2.2. Case I. The following lemma will be the key tool for proving uniqueness and stability when n ≤ (r − 1)m.
Lemma 4. Given m and n ≤ (r − 1)m, let G be an n-vertex graph and M a subset of V (G) with |M | = m such that no copy of K r in G uses vertices of M . Suppose that there are sets P 1 , . . . , P k of sizes p 1 , . . . , p k in G such that the following holds for all Proof. We first establish some simple bounds on the number of edges in G. Each P i contains p i 2 edges. By the maximality of P 1 , . . . , P k we have Putting these estimates together we obtain Observe that the right hand side of (3) defines a function, which we denote g n (p 1 , . . . , p k ), whose domain is the set of tuples (of any length k) of nonnegative integers. In particular we allow k = 0, when (3) gives g n () = t r−1 (n). We now give two equalities relating values of g n . As a preparatory step, observe that for any n ′ we have Now suppose that k ≥ 1. If p k > r then plugging (4) (with n ′ = n − p = n − k ℓ=1 p ℓ ) into the definition of g n in (3) we obtain Similarly, if p k = r then (5) implies We note that our condition n ≤ (r − 1)m implies that m − n−p r−1 − 1 > 0. Applying repeatedly both (6) and (7) we obtain which together with e(G) ≤ g n (p 1 , . . . , p k ) and g n () = t r−1 (n) yields the desired bound on e(G).
We are now ready to prove Case I.
Proofs of Theorems 2 and 3, Case I. Let G be an n-vertex graph and M a subset of V (G) of size m, where n ≤ (r − 1)m, such that no K r of G intersects M . We iteratively find vertex disjoint cliques P 1 , . . . , P k of sizes p 1 , . . . , p k with at least r vertices as follows. Suppose that for some i, the cliques P 1 , . . . , P i−1 have already been defined. Let P i be an arbitrary maximum clique on at least r vertices in the graph G − j<i P j . We set k := i − 1 and terminate if no such clique exists. Let p := k ℓ=1 p ℓ . Now G, M and P 1 , . . . , P k satisfy the conditions of Lemma 4, so we have We first prove Theorem 2. We distinguish two cases. First, G contains no copy of K r . In this case Turán's theorem guarantees that e(G) ≤ t r−1 (n) with equality if and only if G = T r−1 (n).
Second, G contains at least one copy of K r . In this case, there is at least one term in the double sum in (8) (since P 1 exists) and the smallest of the summands is that with i = 1 and j = p 1 − r, i.e., Since n ≤ (r − 1)m, we have m ≥ n r−1 and hence the smallest summand is at least 1. It follows that e(G) < t r−1 (n) and so G is not extremal. This proves (a ) and (b ).
By definition of the sets P i the graph G ′ is K r -free. Therefore, by Theorem 1 the graph G ′ is ε * -close to T r−1 (v(G ′ )). It follows that G is ε * v(G ′ ) 2 + γ 1 n 2 /2 -near to T r−1 (n), and thus by (9) that G is ε-close to T r−1 (n) as required.
2.3. Case II. We first state three lemmas which we will use to prove Theorems 2 and 3 in Case II. Note that the first two of these lemmas do not require the condition n > (r − 1)m. The first lemma asserts that every graph G with no K r intersecting M can easily be modified such that each vertex outside M has high degree. (c ) Either e(G ′ ) > e(G) + µ 2 n 2 , or G ′ is µn 2 -near to G (without relabelling vertices).
Proof. We obtain G ′ from G by repeating the following procedure until conclusion (d ) is satisfied. If there exists a vertex v ∈ V (G) \ M with degree smaller than n − m − µn − 1, delete all edges containing v and insert all edges from v to V (G) \ M ∪ {v} .
Observe that at each step, we add at least µn edges to the graph, and edit at most n edges. It follows that the algorithm terminates, and thus conclusions (b ) and (d ) get satisfied. Clearly, the resulting graph G also satisfies (a ). Furthermore, if the procedure is repeated more than µn times, then e(G ′ ) − e(G) > µ 2 n 2 , while otherwise the number of edits is at most µn 2 , so conclusion (c ) is satisfied.
The next lemma states that there are few vertices which have big degree in G and many neighbours in M . Proof. Let x 1 , . . . , x k be the vertices of a maximum clique in G[X]. For each i ∈ [k], let s i be the number of non-neighbours of x i in X (including x i itself). Because x 1 , . . . , x k is a maximum clique, every vertex of X is a non-neighbour of at least one x i , and therefore we have s 1 + . . . + s k ≥ |X|.
Observe that x i has at most n − m − s i neighbours outside M . Hence, by definition of X and since deg(x i ) ≥ n − m − ν 2 n the vertex x i has at least max νn, s i − ν 2 n neighbours in M . On the other hand, no vertex of M is adjacent to more than r − 2 of the vertices x 1 , . . . , x k , or there would be a copy of K r intersecting M . It follows that (r − 2)|M | ≥ kνn and The final preparatory lemma asserts that T r−1 (n, m) is closed under certain local modifications.
Lemma 7. Suppose that n > (r − 1)m. Let G 1 ∈ T r−1 (n, m) be a graph in which no K r intersects the m-set M ⊆ V (G 1 ), and let v ∈ V (G 1 ) \ M be a vertex whose neighbourhood in G 1 is V (G 1 ) \ M ∪ {v} . Delete all edges incident to v and insert n − m − 1 edges, of which at least one goes to M . If there is no copy of K r intersecting M in the modified graph G 2 , then G 2 ∈ T r−1 (n, m).
Proof. Recall that since G 1 is in T r−1 (n, m), it contains a copy of the graph T r−1 ((r − 1)m) with colour classes V 1 , . . . , V r−1 which covers M , but which does not cover v because each of its vertices is either in or adjacent to M in G 1 . The same sets V 1 , . . . , V r−1 continue to induce a copy of T r−1 ((r − 1)m) in G 2 . Since v has at least one G 2 -neighbour in M , we can let w i be a neighbour of v in M ∩ V i for some i. If v is adjacent to at least one vertex of each set V 1 , . . . , V r−1 , then letting w j be a neighbour of v in V j for each j = i, we obtain a copy of K r in G 2 intersecting M , which is a contradiction. Thus there is j such that v has no neighbours in V j , and since v has degree n − m − 1 it follows that the neighbourhood of v is precisely V (G 1 ) \ V j ∪ {v} . In other words, v has the same neighbourhood as a sporadic vertex in our construction, and we need only to show that there is If such a vertex existed, then v, v ′ and w i together with one vertex in each set V ℓ with ℓ ∈ {i, j} would form a copy of K r intersecting M in G 2 .
We can now prove Case II.
Proof of Theorems 2 and 3, Case II. Let G = (V, E) and M satisfy the conditions of the theorems. First we show that e(G) ≤ t r−1 (n, m), with equality only for graphs in T r−1 (n, m), which will prove Theorem 2.
We apply Lemma 5 to G with µ := 0 to obtain a graph G ′ on V which also has no K r intersecting M , which has e(G ′ ) ≥ e(G) with equality only if G = G ′ , and which is such that every vertex v ∈ V \ M has deg G ′ (v) ≥ n − m − 1. We now apply repeatedly the following further transformation to G ′ to obtain G ′′ . If there exists a vertex v in V \ M whose degree is n − m − 1 and which has a neighbour in M , we delete all edges incident to v, and insert all edges from v to V \ M ∪ {v} . Observe that e(G ′′ ) = e(G ′ ), and G ′′ satisfies the conditions of Lemma 6 with ν := 0. It follows that the set X of G ′′ -neighbours of M in V \ M has size |X| ≤ (r − 2)m. Let X ′ be a subset of V \ M containing X of size exactly (r − 2)m.
Since |X ′ ∪ M | = (r − 1)m, we can now apply Theorem 2 in Case I to conclude that Observe that the vertices in V \(X ′ ∪M ) are all of degree n−m−1 and have no neighbours in M . It follows that e(G ′′ ) ≤ t r−1 (n, m), with equality only if G ′′ ∈ T r−1 (n, m). Since e(G) ≤ e(G ′ ) = e(G ′′ ), we have e(G) ≤ t r−1 (m, n), with equality only if G = G ′ and G ′′ ∈ T r−1 (n, m). It remains only to show that if e(G) = t r−1 (n, m), then the transformation from G = G ′ to G ′′ cannot take a graph outside T r−1 (n, m) to a graph in T r−1 (n, m). Observe that the reverse of this transformation consists exactly of steps satisfying Lemma 7, which therefore asserts that since G ′′ ∈ T r−1 (n, m), so G = G ′ ∈ T r−1 (n, m). This proves Theorem 2.
We apply Lemma 5 to G to obtain a graph G ′ in which no copy of K r intersects M , with e(G ′ ) ≥ e(G), and in which every vertex v ∈ V \ M has deg G ′ (v) ≥ n − m − µn − 1. In particular, we have e(G ′ ) ≤ t r−1 (n, m). Since µ 2 > γ by (11), we must have e(G ′ ) ≤ e(G) + µ 2 n 2 , so by conclusion (c ) of Lemma 5 the graph G ′ is obtained from G by editing at most µn 2 edges. Now since µn ≥ 1 and by (11), we have deg Letting X be the vertices in V \ M with at least νn neighbours in M , we obtain by Lemma 6 that |X| ≤ (1+ν)(r−2)m.
Let H = G ′′ [X ′ ∪M ]. Since there are no edges in G ′′ between V \(X ′ ∪M ) and M , we have .

Concluding remarks
In our main results, we consider forbidden copies of K r that intersect M . An obvious extension would be to forbid copies of K r that intersect M in at least s vertices. We suspect that, at least for small s, similar methods to those used here might give corresponding results also for this setting.
Another possible direction of extending Theorem 2 is to forbid a general fixed r-partite graph H, instead of K r , to touch the set M . The standard regularity method allows one to deduce that the upper bound from Theorem 2 holds even in this case, up to an additive o(n 2 ) term. The Turán graph provides an almost matching lower bound in Case I. The regularity method proves the corresponding counterpart to Theorem 3 in Case I as well. In Case II, however, the graphs in T r−1 (n, m) do not necessarily provide a lower bound. For example, each of the graphs in T 2 (n, m) contains a copy of C 5 touching the set M . It would be interesting to determine the true extremal results in such cases.
Finally, one could ask for a stronger stability result in the spirit of [7]. That is, we want to prove that if e(G) > t r−1 (n) − o(n 2 ) then after deleting o(n) we get a subgraph of a graph from T r−1 (n, m). This can be obtained easily from Theorem 3 as follows. We take the graph G ′ on the vertex set V (G) in T r−1 (n, m) with edit distance less than εn 2 to G guaranteed by Theorem 3. We now remove from V (G) all vertices whose neighbourhoods in G and G ′ do not have symmetric difference less than 2 √ εn. Because G and G ′ are close in edit distance we remove at most √ εn vertices. We further remove vertices V (G) that are either sporadic vertices of G ′ , or that lie in a set V i ∩ Y or V i \ Y of size less than 4r √ εn to obtain the vertex set V ′ , with |V ′ | ≥ 1 − 10r 2 √ ε n. It is now easy to check that if G[V ′ ] is not a subgraph of G ′ [V ′ ] then there is a copy of K r in G[V ′ ] touching M , a contradiction.

Acknowledgements
This paper was finalised during the Midsummer Combinatorial Workshop 2014 in Prague. We would like to thank the organisers for their hospitality.