New short proofs to some stability theorems

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Abstract

We present new short proofs to both the exact and the stability result of two extremal problems. The first result is about the extension of Turán’s theorem to hypergraphs, and the second result is about cancellative hypergraphs. Our proofs are concise and straightforward, but give a sharper version of stability theorems to both problems.

Introduction

For a set V and an integer r2, we use Vr to denote the collection of all r-subsets of V. An r-graph H is a collection of r-subsets of some ground set V(H), which is called the vertex set of H. The r-sets in H are called edges of H. We identify an r-graph H with its edge set and use |H| to denote the number of edges in H. For a family F of r-graphs we say H is F-free if it does not contain any r-graph in F as a subgraph. The Turán number ex(n,F) of F is the maximum number of edges in an n-vertex F-free r-graph, and F is called non-degenerate if the Turán density1 π(F)limnex(n,F)nr is not 0.

Determining, even asymptotically, the value of ex(n,F) for non-degenerate families F of r-graphs with r3 is known to be notoriously hard in general. Many families F have the property that there is a unique r-graph on n vertices with ex(n,F) edges, and every F-free hypergraph with close to ex(n,F) edges is also structurally close to this unique r-graph. Such a property of F is called stability. It is both an interesting property of F itself and also an extremely useful tool in determining the exact value of ex(n,F). The Turán numbers of many families F have been determined by using this method, and we refer the reader to a survey by Keevash [12] for results before 2011.

In this paper, we focus on the stability properties of two extremal problems. The first result is the extension of Turán’s Theorem to hypergraphs, which was firstly studied by Mubayi [16].

By [n] we denote the set {1,,n}. Let V1V be a partition of [n] with each part of size either n or n. The r-graph Tr(n,) is the collection of all r-sets that intersect each Vi on at most one vertex. Further we denote by tr(n,) the number of edges in Tr(n,). The family K+1(r) is the collection of all r-graphs F with at most +12 edges such that for some (+1)-set S every pair {x,y}S is covered by an edge in F. Notice that T2(n,) is just the ordinary Turán graph, and K+1(2) is the set containing only the ordinary complete graph K+1.

In [16] Mubayi proved both the exact and stability results for K+1(r)-free r-graphs.

Theorem 1.1 Mubayi [16]

Let n1 and r2 be integers. Then ex(n,K+1(r))=tr(n,).

Moreover, Mubayi showed that Tr(n,) is the unique maximum K+1(r)-free r-graph on n vertices.

Theorem 1.2 Stability; Mubayi [16]

Fix r2. For every δ>0 there exist an ϵ>0 and an n0 such that the following holds for all nn0. Let H be an n-vertex K+1(r)-free r-graph with at least (1ϵ)tr(n,) edges. Then the vertex set of H has a partition V1V such that all but at most δnr edges have at most one vertex in each Vi.

Mubayi does not give an explicit relation between ϵ and δ, but our proof will show that it suffices to choose δ=ϵ(r2)!. We note that the inductive proof on +r given by de Oliveira Contiero, Hoppen, Lefmann, and Odermann [4] also shows that a linear dependency between δ and ϵ, namely δ=(1+o(1))rϵ, is sufficient2. We do not intend to find the optimal relation between δ and ϵ, but it would be interesting to see whether our proof can be used to improve the relation between δ and ϵ substantially.

The second problem we consider in this paper is about cancellative hypergraphs, which was firstly studied by Bollobás [3] and later by Keevash and Mubayi [13].

A hypergraph H is called cancellative if it does not contain three distinct sets A,B,C with ABC. Note that an ordinary graph G is cancellative iff it contains no triangles (i.e. K3), and Mantel’s theorem states that the maximum size of a cancellative graph is uniquely achieved by the Turán graph T2(n,2). Motivated by Mantel’s theorem, in the 1960s, Katona raised the question of determining the maximum size of a cancellative 3-graph and conjectured that the maximum size of a cancellative 3-graph is achieved by T3(n,3). Katona’s conjectured was proved by Bollobás in [3].

Theorem 1.3 Bollobás [3]

A cancellative 3-graph on n vertices has at most t3(n,3) edges.

Bollobás also showed that T3(n,3) is the unique maximum cancellative 3-graph on n vertices. A new proof of Bollobás’s result was given by Keevash and Mubayi [13] who further proved the following stability result.

Theorem 1.4 Stability; Keevash and Mubayi [13]

For every δ>0 there exist ϵ>0 and n0 such that the following holds for all nn0. Every n-vertex cancellative 3-graph with at least (1ϵ)t3(n,3) edges has a partition of its vertex set as V1V2V3 such that all but at most δn3 edges of H has one vertex in each Vi.

In their proof they also gave an explicit relation between ϵ and δ, which is ϵ<272×1024δ6. Our proof will show that it suffices to choose ϵ=δ100.

The rest of this paper is organized as follows. In Section 2 we prove Theorem 1.1 for divides n and Theorem 1.2. In Section 3 we prove Theorem 1.3 for 3 divides n and Theorem 1.4. In Section 4 we present a short proof to the stability of a generalized Turán problem in graph theory. In Section 5 we present a brief discussion about the relationship between ϵ and δ and some related results.

Section snippets

Proofs of Theorems 1.1 and 1.2

In this section we prove Theorem 1.1 for divides n and Theorem 1.2. Our proofs are based on two results, and the first result is a stability theorem for K+1-free graphs.

Theorem 2.1

Füredi [9]

Let t0 be an integer, and let G be an n-vertex K+1-free graph with t2(n,)t edges. Then G contains an -partite subgraph G with at least t2(n,)2t edges.

The second result describes a relationship between the number of copies of Kr1 and Kr2 in a K+1-free graph, where r1 and r2 are two positive integers less than +1.

Theorem 2.2

Fisher and Ryan [7]

Let G

Proofs of Theorems 1.3 and 1.4

In this section we prove Theorem 1.3 for 3 divides n and Theorem 1.4. Let us first present some preliminary definitions and results.

The following inequality will be used intensively in our proofs.

Lemma 3.1

Jensen’s Inequality [10]

Suppose that f:IR is a convex function on some interval I and x1,,xnI. Then i[n]f(xi)nfi[n]xin.

Let H be an r-graph on [n]. A set I[n] is independent if every edge in H contains at most one vertex in I. For every nonempty set S[n] define the link LH(S) of S in H as LH(S)=AH:A{s}H,sS,and

Applications to the generalized Turán problems

In this section we present some applications of Theorem 2.2 for the generalized Turán problems.

Let T and H be two ordinary graphs. Denote by ex(n,T,H) the maximum possible number of copies of T in an ordinary H-free graph on n vertices. The function ex(n,T,H) is called the generalized Turán number.

Fix r3. In [5] Erdős proved that ex(n,Kr,K+1)tr(n,). A similar argument as in the proof of Theorem 1.2 also gives the following stability result to ex(n,Kr,K+1). For the sake of completeness

Concluding remarks

We showed that a linear dependence between δ and ϵ is sufficient for Theorem 1.2, Theorem 1.4, Theorem 4.1, Theorem 4.2, and in [9] Füredi showed that a linear dependence between δ and ϵ is also sufficient for Theorem 2.1. It seems to be an interesting problem in general to determine the exact relationship between ϵ and δ in these stability theorems, and we refer the reader to [2], [6], [14] for related results on this topic.

Acknowledgments

We thank Dhruv Mubayi for his suggestions that have greatly improved the presentation of this paper, and we also thank József Balogh for alerting us on Ref. [6]. We are very grateful to the referee for many helpful suggestions.

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