aa r X i v : . [ m a t h . C O ] A p r New Short Proofs to Some Stability Theorems
Xizhi Liu ∗ April 3, 2019
Abstract
We present new short proofs to both the exact and the stability results of twoextremal problems. The first one is the extension of Tur´an’s theorem in hypergraphs,which was firstly studied by Mubayi [1]. The second one is about the cancellativehypergraphs, which was firstly studied by Bollob´as [2] and later by Keevash andMubayi [3]. Our proofs are concise and straightforward, but give a sharper version ofstability theorems to both problems.
Let H be an n -vertex r -graph and let F be a family of r -graphs. H is F -free if it does notcontain any r -graph in F as a subgraph. The Tur´an number ex ( n, F ) is the maximumnumber of edges in an n -vertex F -free r -graph. F is called non-degenerate if the Tur´andensity π ( F ) := lim n →∞ ex ( n, F ) / (cid:0) nr (cid:1) is not 0.Determining, even asymptotically, the value of ex ( n, F ) for general non-degenerate r -graphs F with r ≥ F have the property that there is a unique extremal family attain the value ex ( n, F ),and any F -free hypergraph with close to ex ( n, F ) edges is also structurally close to theextremal family. This property of F is called stability . It is both an intersecting propertyof F and also an extremely useful tool in determining the value of ex ( n, F ). The Tur´annumbers for many families F has been determined by using this method, and we refer thereader to a survey by Keevash [4] for results before 2011.In the present paper, we mainly focus on the stability properties for two extremalproblems. The first one is the extension of Tur´an’s theorem in hypergraphs, and it wasfirstly studied by Mubayi [1].Let V ∪ ... ∪ V ℓ be a partition of [ n ] with each part of size either ⌊ n/ℓ ⌋ or ⌈ n/ℓ ⌉ . T r ( n, ℓ ) is the family of all r -sets that intersect each V i in at most one vertex. Let t r ( n, ℓ )denote the number of edges in T r ( n, ℓ ). K ( r ) ℓ +1 is the family of all r -graphs F with at most (cid:0) ℓ +12 (cid:1) edges such that for some ( ℓ + 1)-set S every pair x, y ∈ S is covered by an edge of F . Notice that T ( n, ℓ ) is just the ordinary Tur´an graph, and K (2) ℓ +1 is just the ordinarycomplete graph on ℓ + 1 vertices, which is also denoted by K ℓ +1 .In [1] Mubayi proved both the exact and stability result for K ( r ) ℓ +1 -free r -graphs. Theorem 1.1 (Mubayi, [1]) . Let n, ℓ, r ≥ . Then ex ( n, K ( r ) ℓ +1 ) = t r ( n, ℓ ) and T r ( n, ℓ ) is the unique maximum K ( r ) ℓ +1 -free r -graph on n vertices. ∗ Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, IL,60607 USA. Email: [email protected] heorem 1.2 (Stability; Mubayi, [1]) . Fix l ≥ r ≥ . For every δ > , there exists an ǫ > and an n such that the following holds for all n ≥ n . Let G be an n -vertex K ( r ) ℓ +1 -free r -graph with at least (1 − ǫ ) t r ( n, ℓ ) edges. Then the vertex set of G has a partition V ∪ . . . ∪ V ℓ such that all but at most δn r edges have at most one vertex in each V i . Note that in [1] Mubayi did not give an explicit relation between ǫ and δ , but our proofwill show that it suffices to choose ǫ = ( r − δ . Also, note that in [5] Contiero, Hoppen,et al. also proved a linear dependence between δ and ǫ by induction on ℓ + r , but ourproof is different and much shorter.The second one is about the cancellative hypergraphs, and it was firstly studied byBollob´as [2] and later by Keevash and Mubayi [3].A hypergraph H is called cancellative if it does not contain three distinct sets A, B, C with A △ B ⊂ C . Note that an ordinary graph G is cancellative iff it does not contain atriangle (i.e. K ), and Mantel’s theorem states that the maximum size of a cancellativegraph is uniquely achieved by T ( n, T ( n, Theorem 1.3 (Bollob´as, [2]) . A cancellative -graph on n vertices has at most t ( n, edges, with equality only for T ( n, . In [3] a new proof of Bollob´as’ result was given by Keevash and Mubayi, and they alsoproved a stability theorem for cancellative 3-graphs.
Theorem 1.4 (Stability; Keevash and Mubayi, [3]) . For any δ > there exists ǫ > and n such that the following holds for all n ≥ n . Any n -vertex cancellative -graph with atleast (1 − ǫ ) t ( n, edges has a partition of vertex set as [ n ] = V ∪ V ∪ V such that allbut at most δn edges of H has one vertex in each V i . In their proof they also gave an explicit relation between ǫ and δ , which is ǫ < / × − δ . Our proof will show that it suffices to choose ǫ = δ/ ǫ and δ . Let H be an r -graph on [ n ]. The size of H is the number of edges in H , which is denotedby | H | . I ⊂ [ n ] is an independent set if every edge in H contains at most one vertex of I .The shadow of H , denoted by ∂H , is defined as ∂H = (cid:26) A ∈ (cid:18) [ n ] r − (cid:19) : ∃ B ∈ H such that A ⊂ B (cid:27) For every nonempty set S ⊂ [ n ], define the link L ( S ) of S in H to be L ( S ) = { A ∈ ∂H : A ∪ { s } ∈ H, ∀ s ∈ S } For convenience, we use L ( u ) to represent L ( { u } ), and use L ( u, v ) to represent L ( { u, v } ).Note that in our proof L ( u, u ) also represents L ( { u } ).2et T ∈ ∂H , the neighborhood of T in H is defined as N ( T ) = { v ∈ [ n ] : T ∪ { v } ∈ H } and the degree of T is d ( T ) = | N ( T ) | . It follows from an easy double counting that X T ∈ ∂H d ( T ) = 3 | H | The edge set of an ordinary graph G can be viewed as a family of unordered pairs. Tokeep the calculations in our proof simply, we define an auxiliary family ~G of order pairsas ~G = { ( u, v ) : { u, v } ∈ G } . Note that if { u, v } ∈ G , then ( u, v ) and ( v, u ) are bothcontained in ~G and hence we have | ~G | = 2 | G | . Let N be a set, we use N to denotethe cartesian product N × N , which is also the collection of all ordered pairs ( u, v ) with u, v ∈ N . Here u and v might be the same.Our proof of theorems 1.1 and 1.2 is based on two results. The first one is the stabilityof K ℓ +1 -free graphs. Theorem 2.1 (F¨uredi, [6]) . Let t ≥ and let G be an n -vertex K ℓ +1 -free graph with t ( n, ℓ ) − t edges. Then G contains an ℓ -partite subgraph G ′ with at least t ( n, ℓ ) − t edges. The second one describes an relation between the number of copies of K r and K r ina K ℓ +1 -free graph, where r and r are two positive integers less that ℓ + 1. Theorem 2.2 (Fisher and Ryan, [7]) . Let G be an n -vertex K ℓ +1 -free graph. For every i ∈ [ ℓ ] , let k i denote the number of copies of K i in G . Then k ℓ (cid:0) ℓℓ (cid:1) ! ℓ ≤ k ℓ − (cid:0) ℓℓ − (cid:1) ! ℓ − ≤ ... ≤ k (cid:0) ℓ (cid:1) ! ≤ k (cid:0) ℓ (cid:1) ! (1)To prove theorems 1.3 and 1.4 we first present two simply properties of cancellative3-graphs. Lemma 2.3.
Let H be a cancellative -graph, and v is a vertex in H . Then the link graph L ( v ) is triangle-free.Proof. Suppose { x, y, z } is a triangle in L ( v ). Then { v, x, y } , { v, x, z } , { v, y, z } are allcontained in H , but { v, x, y }△{ v, x, z } = { y, z } ⊂ { v, y, z } which is a contradiction. Therefore, L ( v ) is triangle-free. Lemma 2.4.
Let H be a cancellative -graph, and T ∈ ∂H . Then N ( T ) is an independentset.Proof. Let u, v ∈ N ( T ) and let A = { u } ∪ T and A = { v } ∪ T . Note that A and A are contained in H . Since A △ A = { u, v } and by assumption there is no edge in H containing { u, v } . Therefore, N ( T ) is an independent set.In the proof of theorem 1.4 we need the following lemma, which is essentially thestability of triangle-free graphs. For completeness we include its proof here.Let G be an ordinary graph and let v be a vertex in G . We use N G ( v ) to denote theneighborhood of v in G , and use d G ( v ) to denote the degree of v in G .3 emma 2.5. Let G be a triangle-free graph on [ n ] with at least (1 − ǫ )( n/ edges.Then G contains two vertices v and v such that N G ( v ) and N G ( v ) are disjoint and | N G ( v ) | + | N G ( v ) | ≥ (1 − ǫ ) n .Proof. Since G is triangle-free. So N G ( u ) and N G ( v ) are disjoint for all edge uv in G .Therefore, it suffices to find an edge uv in G such that d G ( u )+ d G ( v ) ≥ (1 − ǫ ) n . Combiningan easy counting argument with the Jensen Inequality we obtain X uv ∈ E ( G ) ( d G ( u ) + d G ( v )) = X v ∈ V ( G ) d G ( v ) ≥ (cid:16)P v ∈ V ( G ) d G ( v ) (cid:17) n = 4 e ( G ) n It follows from an averaging argument that there exists an edge uv with d G ( u ) + d G ( v ) ≥ e ( G ) /n ≥ (1 − ǫ ) n . Let H be a K ( r ) ℓ +1 -free r -graph on [ n ]. Define an auxiliary graph G = (cid:26) A ∈ (cid:18) [ n ]2 (cid:19) : ∃ B ∈ H such that A ∈ B (cid:27) Let us state two easy facts about the relation between H and G without proof. Lemma 3.1. (a). H is K ( r ) ℓ +1 -free iff G is K ℓ +1 -free.(b). The number of edges in H is at most the number of copies of K r in G . Proof of theorem 1.1:
Combining lemma 3 . | H | ≤ (cid:0) ℓr (cid:1) (cid:0) nℓ (cid:1) r . This proves theorem 1.1 for the case ℓ divides n , and we omit the proof of theother case. Proof of theorem 1.2:
Choose ǫ = ( r − δ , and let n be sufficiently large. Byassumption we have | H | ≥ (1 − ǫ ) t r ( n, ℓ ) ≥ (1 − ǫ ) (cid:0) ℓr (cid:1) ( n/ℓ ) r . Combining lemma 3 . e in G satisfies e ≥ (1 − ǫ ) /r (cid:18) ℓ (cid:19) (cid:16) nℓ (cid:17) ≥ (1 − ǫ ) (cid:18) ℓ (cid:19) (cid:16) nℓ (cid:17) ≥ (1 − ǫ ) t ( n, ℓ )Therefore, by theorem 2.1, G has a vertex set partition V ∪ ... ∪ V ℓ such that all but atmost 2 ǫt ( n, ℓ ) edges of G have at most one vertex in each V i . It follows that all but atmost 2 ǫt ( n, ℓ ) (cid:0) nr − (cid:1) ≤ ǫn r / ( r − ≤ δn r edges of H have at most one vertex in each V i .This completes the proof of theorem 1.2. The most improtant step in this section is building an relation between H and ∂H , whichis equation (2). Proof of theorem 1.3:
Let us count the number of ordered pairs ( u, v ) in [ n ] \ −→ ∂H . Bylemma 2.4, if { u, v } is contained in N ( e ) for some e ∈ ∂H , then { u, v } can not be containedin ∂H . Since every set S ⊂ [ n ] is contained in exactly | L ( S ) | sets in { N ( T ) : T ∈ ∂H } .Therefore, we have X T ∈ ∂H X ( u,v ) ∈ N ( T ) | L ( u, v ) | ≤ n − | ∂H | (2)4ombining lemma 2.3 with Mantel’s theorem we obtain that | L ( u, v ) | ≤ ( n − d ( T )) / u, v ) ∈ [ n ] . It follows from (2) that X T ∈ ∂H (cid:18) d ( T ) n − d ( T ) (cid:19) ≤ n − | ∂H | Since ( x/ ( n − x )) is convex for x ∈ [0 , n ], it follows from Jensen’s inequality that4 (cid:18) | H | / | ∂H | n − | H | / | ∂H | (cid:19) | ∂H | ≤ n − | ∂H | Now let z = | H | / | ∂H | n − | H | / | ∂H | . Then (4) implies | ∂H | ≤ n z + 1)Substitute | H | = zn z +1) | ∂H | into the equation above we obtain | H | ≤ z z + 1)(2 z + 1) n Since the maximum of z z +1)(2 z +1) is 1 /
27. Therefore, we have | H | ≤ (cid:0) n (cid:1) . This provestheorem 1.3 for the case 3 divides n , and we omit the proof of the other case.Choose ǫ = δ/ H be a cancellative 3-graph on [ n ] with at least (1 − ǫ ) t ( n, > (1 − ǫ )( n/ edges. Before we prove theorem 1.4, let us present a lemma follows fromequation (2). Lemma 4.1.
There exists T ∈ ∂H such that X ( u,v ) ∈ N ( T ) | L ( u, v ) | ≥ (1 − ǫ ) d ( T ) (cid:18) n − d ( T )2 (cid:19) (3) Proof.
Suppose that (3) is false for all T ∈ ∂H . Since 1 /x is convex for x >
0, it followsfrom Jensen’s inequality that X ( u,v ) ∈ N ( T ) | L ( u, v ) | ≥ d ( T ) P ( u,v ) ∈ N ( T ) | L ( u, v ) | /d ( T ) > d ( T )(1 − ǫ ) ( n − d ( T )) (4)Substitute (4) into (2) we obtain X T ∈ ∂H d ( T )(1 − ǫ ) ( n − d ( T )) ≤ n − | ∂H | Similar argument as in the proof of theorem 1.3 yields | H | ≤ z z + 1) (cid:16) z − ǫ + 1 (cid:17) n By assumption we have | H || ∂H | ≥ − ǫ )( n/ n / ≥ /
9. Therefore, we may assume that z > / z − ǫ + 1 > z +11 − ǫ . So we obtain z z + 1) (cid:16) z − ǫ + 1 (cid:17) < (1 − ǫ ) z z + 1) (2 z + 1) ≤
127 (1 − ǫ )This implies that | H | < (1 − ǫ ) (cid:0) n (cid:1) < (1 − ǫ ) t ( n, roof of theorem 1.4: Choose T ∈ ∂H such that (3) holds for T . Let V ′ = N ( T ).By Pigeonhole principle, there exists a pair ( u, v ) ∈ N ( T ) such that | L ( u, v ) | ≥ (1 − ǫ ) (( n − d ( T )) / . Let L denote the graph L ( u, v ) and let U denote the vertex set[ n ] \ N ( T ). Combining lemma 2.4 with lemma 2.5 we know that there exist two vertices x and y in U such that N L ( x ) and N L ( y ) are disjoint and N L ( x ) + N L ( y ) ≥ (1 − ǫ )( n − d ( T )). Let V = N L ( x ) and V = N L ( y ). Note that N L ( x ) = N ( ux ) and N L ( y ) = N ( uy )and hence V and V are independent sets in H .Now we have independent sets V ′ , V and V , and | V ′ | + | V | + | V | ≥ d ( T ) + (1 − ǫ )( n − d ( T )) > n − ǫn . Let V = [ n ] \ ( V ∪ V ). The number of edges in H that hasat least two vertices in some V i is at most (cid:0) ǫn (cid:1) + (cid:0) ǫn (cid:1)(cid:0) n (cid:1) + (cid:0) ǫn (cid:1)(cid:0) n (cid:1) < ǫn = δn .This completes the proof of theorem 1.4. In this section we present some applications of equation (1) in the generalized Tur´anproblems.Let T and H be two ordinary graphs. Let ex ( n, T, H ) denote the maximum possiblenumber of copies of T in an ordinary H -free graph on n vertices. The function ex ( n, T, H )is called the generalized Tur´an number.Fix ℓ ≥ r ≥
3. In [8] Erd˝os proved that ex ( n, K r , K ℓ +1 ) ≤ t r ( n, ℓ ). Actually a similarargument as in the proofs of theorems 1.1 and 1.2 also gives an exact and stability resultto ex ( n, K r , K ℓ +1 ). Here we state the stability result without proof. Theorem 5.1.
Fix ℓ ≥ r ≥ , and δ > . Then there exists an ǫ > and an n such thatthe following holds for all n ≥ n . If G is an n -vertex K ℓ +1 -free graph containing at least (1 − ǫ ) (cid:0) ℓr (cid:1) t r ( n, ℓ ) copies of K r , then G has a vertex set partition V ∪ . . . ∪ V ℓ such that allbut at most δn edges have at most one vertex in each V i . Note that our proof implies that it suffices to choose ǫ = δ .In [9] Alon and Shikhelman studied the function ex ( n, T, H ) for other combinationsof T and H . In particular they proved that ex ( n, K r , H ) = (1 + o (1)) t r ( n, ℓ ) holds forevery graph H with chromatic number χ ( H ) = ℓ + 1. Later their result was improved byMa and Qiu [10], who proved that ex ( n, K r , H ) = t r ( n, ℓ ) + biex( n, H ) · Θ( n r − ), wherebiex( n, H ) is the Tur´an number of the decomposition family of H . Moreover they proveda stability result for ex ( n, K r , H ). Theorem 5.2 (Ma and Qiu, [10]) . Fix ℓ ≥ r ≥ , and δ > . For every graph H withchromatic number ℓ + 1 , there exists an ǫ > and an n such that the following holds forall n ≥ n . If G is an n -vertex H -free graph containing at least (1 − ǫ ) (cid:0) ℓr (cid:1) t r ( n, ℓ ) copiesof K r , then G has a vertex set partition V ∪ . . . ∪ V ℓ such that all but at most δn edgeshave at most one vertex in each V i . Here we present a short proof to theorem 5.2 using theorem 5.1 and the RemovalLemma, and our proof implies that it is suffices to choose ǫ = δ/ Theorem 5.3 (Removal Lemma, e.g. see [6], [11]) . Let H be a graph with chromaticnumber ℓ + 1 . For every δ > there exists an n such that the following holds for all n ≥ n . Every n -vertex H -free graph G can be made K ℓ +1 -free by removing at most δn edges. roof of Theorem 5.2: Let n be sufficiently large. Choose ǫ = δ/
3. Let G be an n -vertex H -free graph containing at least (1 − ǫ ) (cid:0) ℓr (cid:1) t r ( n, ℓ ) copies of K r . By the RemovalLemma, G contains a K ℓ +1 -free subgraph G ′ with at least e ( G ) − ǫn /ℓ r edges. Since everyedge e in G is contained in at most (cid:0) nr − (cid:1) copies of K r in G . Therefore, the number ofcopies of K r in G ′ is at least (1 − ǫ ) (cid:0) ℓr (cid:1) t r ( n, ℓ ). By theorem 5.1, G ′ has a vertex partition V ∪ . . . ∪ V ℓ such that all but at most 2 ǫn edges in G ′ have at most one vertex in each V i . Therefore, all but at most 3 ǫn edges in G have at most one vertex in each V i . Note that we showed that a linear dependence between δ and ǫ is sufficient for Theorems1.2, 1.4, 5.1 and 5.2, and in [6] F¨uredi showed that a linear dependence is also sufficientfor Theorem 2.1. So one might wondering if the linear dependence between δ and ǫ istight (up to a constant) for the stability theorems above. In other words, if there exists anabsolute constant C > ǫ > δ ≥ Cǫ .We did not try to answer the question above in full generality, but our example belowof K -free graphs shows that the answer seems to be negative.Fix ǫ >
0. Let G = ( V, E ) be an n -vertex K -free graph with (1 / − ǫ ) n edges. Let V ∪ V be a partition of V such that the number of edges in the bipartite graph G [ V , V ]is maximum. Define the set of bad edges B and the set of missing edges M as following. B = { uv ∈ E ( G ) : u, v ∈ V i for some i ∈ { , }} and M = { uv E ( G ) : u ∈ V and v ∈ V } Therefore, in order to make G bipartite one has to remove all edges in B .Assume that | B | = δn . Let B = B ∩ (cid:0) V (cid:1) be the set of bad edges contained in V .Without lose of generality we may assume that | B | ≥ δn / v ∈ V , let N ( v ) be the neighborhood of v in V , and let d ( v ) = | N ( v ) | . Let N ( v ) be the neighborhood of v in V , and let d ( v ) = | N ( v ) | . By themaximality of the partition V ∪ V , we know that d ( v ) ≥ d ( v ) since otherwise one canmove v from V to V to get a larger bipartite subgraph of G . Also we know that there isno edge between N ( v ) and N ( v ) since G is K -free.Now let ∆ = max { d ( v ) : v ∈ V } . Case 1: ∆ ≥ δ / n . Then choose v ∈ V of maximum degree ∆. Since there is noedge between N ( v ) and N ( v ). Therefore, | M | ≥ (∆ n ) ≥ δ / n . On the other hand,we have | M | ≤ ǫn + δn . So δ / ≤ ǫ + δ which implies that lim ǫ → δ/ǫ = 0. Case 2: ∆ < δ / n . Using a greedy strategy one can choose a matching M with atleast (cid:0) δn / (cid:1) / (cid:0) δ / n (cid:1) = δ / n/ B . Let u v , ..., u m v m be the edges in M .Since G is K -free. Therefore, we have d ( u i ) + d ( v i ) ≤ | V | and hence | M | ≥ m X i =1 (2 | V | − d ( u i ) − d ( v i )) ≥ m | V | ≥ δ / n × n δ / n Similarly we obtain that lim ǫ → δ/ǫ = 0.Our example above shows that for K -free graphs there is no absolute constant C > δ/ǫ ≥ C holds for all ǫ >
0. 7
Acknowledgement
The author is very grateful to Dhruv Mubayi for his suggestions that have greatly improvedthe presentation.
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