New short proofs to some stability theorems
Introduction
For a set and an integer , we use to denote the collection of all -subsets of . An -graph is a collection of -subsets of some ground set , which is called the vertex set of . The -sets in are called edges of . We identify an -graph with its edge set and use to denote the number of edges in . For a family of -graphs we say is -free if it does not contain any -graph in as a subgraph. The Turán number of is the maximum number of edges in an -vertex -free -graph, and is called non-degenerate if the Turán density1 is not .
Determining, even asymptotically, the value of for non-degenerate families of -graphs with is known to be notoriously hard in general. Many families have the property that there is a unique -graph on vertices with edges, and every -free hypergraph with close to edges is also structurally close to this unique -graph. Such a property of is called stability. It is both an interesting property of itself and also an extremely useful tool in determining the exact value of . The Turán numbers of many families 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 we denote the set . Let be a partition of with each part of size either or . The -graph is the collection of all -sets that intersect each on at most one vertex. Further we denote by the number of edges in . The family is the collection of all -graphs with at most edges such that for some -set every pair is covered by an edge in . Notice that is just the ordinary Turán graph, and is the set containing only the ordinary complete graph .
In [16] Mubayi proved both the exact and stability results for -free -graphs.
Theorem 1.1 Mubayi [16] Let and be integers. Then .
Moreover, Mubayi showed that is the unique maximum -free -graph on vertices.
Theorem 1.2 Stability; Mubayi [16] Fix . For every there exist an and an such that the following holds for all . Let be an -vertex -free -graph with at least edges. Then the vertex set of has a partition such that all but at most edges have at most one vertex in each .
Mubayi does not give an explicit relation between and , but our proof will show that it suffices to choose . We note that the inductive proof on given by de Oliveira Contiero, Hoppen, Lefmann, and Odermann [4] also shows that a linear dependency between and , namely , is sufficient2. We do not intend to find the optimal relation between and , but it would be interesting to see whether our proof can be used to improve the relation between and 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 is called cancellative if it does not contain three distinct sets with . Note that an ordinary graph is cancellative iff it contains no triangles (i.e. ), and Mantel’s theorem states that the maximum size of a cancellative graph is uniquely achieved by the Turán graph . Motivated by Mantel’s theorem, in the 1960s, Katona raised the question of determining the maximum size of a cancellative -graph and conjectured that the maximum size of a cancellative -graph is achieved by . Katona’s conjectured was proved by Bollobás in [3].
Theorem 1.3 Bollobás [3] A cancellative -graph on vertices has at most edges.
Bollobás also showed that is the unique maximum cancellative -graph on 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 there exist and such that the following holds for all . Every -vertex cancellative -graph with at least edges has a partition of its vertex set as such that all but at most edges of has one vertex in each .
In their proof they also gave an explicit relation between and , which is . Our proof will show that it suffices to choose .
The rest of this paper is organized as follows. In Section 2 we prove Theorem 1.1 for divides and Theorem 1.2. In Section 3 we prove Theorem 1.3 for divides 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 and and some related results.
Section snippets
Proofs of Theorems 1.1 and 1.2
In this section we prove Theorem 1.1 for divides and Theorem 1.2. Our proofs are based on two results, and the first result is a stability theorem for -free graphs.
Theorem 2.1 Let be an integer, and let be an -vertex -free graph with edges. Then contains an -partite subgraph with at least edges.Füredi [9]
The second result describes a relationship between the number of copies of and in a -free graph, where and are two positive integers less than .
Theorem 2.2 Let Fisher and Ryan [7]
Proofs of Theorems 1.3 and 1.4
In this section we prove Theorem 1.3 for divides 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 Suppose that is a convex function on some interval and . Then Jensen’s Inequality [10]
Let be an -graph on . A set is independent if every edge in contains at most one vertex in . For every nonempty set define the link of in as and
Applications to the generalized Turán problems
In this section we present some applications of Theorem 2.2 for the generalized Turán problems.
Let and be two ordinary graphs. Denote by the maximum possible number of copies of in an ordinary -free graph on vertices. The function is called the generalized Turán number.
Fix . In [5] Erdős proved that . A similar argument as in the proof of Theorem 1.2 also gives the following stability result to . For the sake of completeness
Concluding remarks
We showed that a linear dependence between and 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 and is also sufficient for Theorem 2.1. It seems to be an interesting problem in general to determine the exact relationship between and 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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