A generalization of the Kővári-Sós-Turán theorem

02/13/2020 ∙ by Jesse Geneson, et al. ∙ 0

We present a new proof of the Kővári-Sós-Turán theorem that ex(n, K_s,t) = O(n^2-1/t). The new proof is elementary, avoiding the use of convexity. For any d-uniform hypergraph H, let ex_d(n,H) be the maximum possible number of edges in an H-free d-uniform hypergraph on n vertices. Let K_H, t be the (d+1)-uniform hypergraph obtained from H by adding t new vertices v_1, ..., v_t and replacing every edge e in E(H) with t edges e ∪{v_1},..., e ∪{v_t} in E(K_H, t). If H is the 1-uniform hypergraph on s vertices with s edges, then K_H, t = K_s, t. We prove that ex_d+1(n,K_H,t) = O(ex_d(n, H)^1/t n^d+1-d/t). Thus ex_d+1(n,K_H,t) = O(n^d+1-1/t) for any d-uniform hypergraph H with ex_d(n, H) = Θ(n^d-1), which implies the Kővári-Sós-Turán theorem in the d = 1 case. As a corollary, this implies that ex_d+1(n, K_H,t) = O(n^d+1-1/t) when H is a d-uniform hypergraph in which all edges are pairwise disjoint, which generalizes an upper bound proved by Mubayi and Verstraëte (JCTA, 2004). We also obtain analogous bounds for 0-1 matrix Turán problems.

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1 Introduction

The Kővári-Sós-Turán theorem is one of the most famous results in extremal combinatorics [16, 8, 10]. The theorem states that the maximum number of edges in a -free graph of order is . There are multiple known proofs of this theorem, including a standard double-counting proof that uses Jensen’s inequality, as well as a proof that uses dependent random choice and Jensen’s inequality [2].

Past proofs of the Kővári-Sós-Turán theorem have relied on convexity and Jensen’s inequality, or the power mean inequality which is a corollary of Jensen’s inequality. For a student to fully understand the proof, they would need to understand both convexity and Jensen’s inequality, which would require calculus background, or they would need to know an alternative proof of the power mean inequality that avoids using Jensen’s inequality.

In this paper, we prove the Kővári-Sós-Turán theorem without using calculus or Jensen’s inequality. Instead we use a method based on Nivasch’s bounds on Davenport-Schinzel sequences [19] and Alon et al.’s bounds on interval chains [1]. This new proof gives a simple way to teach the proof of the Kővári-Sós-Turán theorem to students with no calculus background, and the same method can be used to prove a generalization of the Kővári-Sós-Turán theorem for uniform hypergraphs.

In [19], Nivasch found upper bounds on the maximum possible lengths of Davenport-Schinzel sequences using two different methods. Both methods gave the same bounds, but the first method was more like the proofs in past papers on Davenport-Schinzel sequences, and the second method was similar to proofs about interval chains in [1]. The second method in [19] was much simpler than the first for proving bounds on Davenport-Schinzel sequences. We imitate the second method here for graph and hypergraph Turán problems.

Let denote the maximum number of edges in an -free -uniform hypergraph on vertices. Let be the -uniform hypergraph obtained from by adding new vertices and replacing every edge in with in . For example, if is the -uniform hypergraph of order with edges, then . Mubayi and Verstraëte [18] proved that when is a -uniform hypergraph in which all edges are pairwise disjoint.

In Section 2, we provide an elementary proof that , giving an alternative proof of the Kővári-Sós-Turán theorem when is the -uniform hypergraph of order with edges. As a corollary, this implies that when is a -uniform hypergraph in which all edges are pairwise disjoint, generalizing the upper bound of Mubayi and Verstraëte. In Section 3, we discuss analogous results about -dimensional 0-1 matrices that can be proved with similar methods.

2 The letter method

An ordered -uniform hypergraph is a -uniform hypergraph with a linear order on the vertices. We define a lettered -uniform hypergraph as the structure obtained from labeling each edge of an ordered -uniform hypergraph with a letter such that two edges can be labeled with the same letter only if they have the same greatest vertex. Given a -uniform hypergraph , we say that a lettered -uniform hypergraph is -free if its underlying -uniform hypergraph is -free.

For any -uniform hypergraph , let denote the maximum possible number of distinct letters in an -free lettered -uniform hypergraph on vertices in which every letter occurs at least times.

The next lemma is analogous to inequalities in [19, 4, 11, 13] and is proved similarly.

Lemma 2.1.

For all positive integers and -uniform hypergraphs , we have .

Proof.

Start with a -uniform -free hypergraph with edges. Order the vertices of arbitrarily. For each vertex in in order from greatest to least, label the unlabeled edges adjacent to in any order with letters , only using each letter exactly times and deleting up to remaining edges adjacent to if does not divide the total number of edges in which is the greatest vertex. Observe that the new lettered hypergraph has at most distinct letters with every letter occurring exactly times, and it is -free. ∎

When combined with Lemma 2.1, the next lemma will complete our proof of the generalization of the Kővári-Sós-Turán theorem. We use Stirling’s bound in the proof of the next lemma, but it is not actually necessary. We explain after the proof how the use of Stirling’s bound can be replaced with an elementary one-sentence argument.

Lemma 2.2.

For and a -uniform hypergraph with , we have .

Proof.

Suppose for contradiction that there exists a -free lettered -uniform hypergraph on vertices with distinct letters in which every letter occurs at least times. Suppose that is sufficiently large so that . Delete edges of until every letter occurs exactly times.

For each -subset of , define to be the number of edges in that contain all of the vertices in and a greater vertex in the ordering. Let be the number of -subsets of with . The number of -tuples of edges in that have the same least vertices is equal to , which is at most , or else would contain a copy of . This follows by the pigeonhole principle, since every -tuple of edges in that have the same least vertices must have different letters on each edge.

Then and , where the last inequality follows from Stirling’s bound. However , a contradiction. ∎

Theorem 2.3.

For fixed and -uniform hypergraph , we have .

The use of Stirling’s bound in Lemma 2.2 may seem to make the proof non-elementary, but it was unnecessary. All we need is that there exists some constant such that for all , and then we can replace each in the last proof with . Proving this for : it is clearly true for , and if we assume that , then we also have , and . Thus the whole proof is elementary.

Corollary 2.4.

If is a -uniform hypergraph in which all edges are pairwise disjoint, then .

Proof.

If is a -uniform hypergraph in which all edges are pairwise disjoint, then , so this bound follows from Theorem 2.3. ∎

The last corollary yields the bound of Mubayi and Verstraëte from [18] when .

3 0-1 matrices

Using the same method, we can get similar bounds for Turán-type problems on -dimensional 0-1 matrices. In order to state these results, we introduce more terminology. We say that -dimensional 0-1 matrix contains -dimensional 0-1 matrix if some submatrix of can be turned into by changing some number of ones to zeroes. Otherwise avoids . For any -dimensional 0-1 matrix , define to be the maximum number of ones in a -dimensional 0-1 matrix of sidelength that avoids .

As with the case of -uniform hypergraphs, most of the past research on the topic of -dimensional 0-1 matrices has focused on when . We mention several results for that have been generalized to higher values of . For example, Klazar and Marcus [15] proved that for every -dimensional permutation matrix , generalizing the result of Marcus and Tardos [17]. Geneson and Tian [14] sharpened this bound by proving that for -dimensional permutation matrices of sidelength , generalizing a result of Fox [6]. Geneson and Tian also proved that for every -dimensional double permutation matrix , generalizing the upper bound in [12].

In order to state the next result, we define to be the -dimensional 0-1 matrix obtained from the -dimensional 0-1 matrix by stacking copies of with the same orientation in the direction of the new dimension. For example if is the matrix of all ones, then is the matrix of all ones.

Theorem 3.1.
  1. For fixed and -dimensional 0-1 matrix , .

  2. For any -dimensional 0-1 matrix with , we have . In particular, for any -dimensional 0-1 matrix with at least two ones such that . Moreover, for any -dimensional 0-1 matrix with at least three ones differing in the first coordinate such that .

Proof.

The upper bounds follow from using the letter method as in the last section. The lower bounds follow from stacking copies of the known lower bound constructions for extremal functions of forbidden and all-ones matrices [3, 5, 9, 10]. ∎

Permutation matrices and double permutation matrices with at least three ones are some examples for which Theorem 3.1 gives sharp bounds on and up to a constant factor [17, 12, 15, 14].

4 Concluding remarks

The standard double-counting method used to prove the Kővári-Sós-Turán theorem can also be used to prove the bounds in Theorem 2.3 and 3.1. We did not include this method since it uses convexity, and it gives the same bounds up to a constant factor as the letter method. Dependent random choice can also be used to obtain the same bounds for uniform hypergraphs up to a constant factor when , and it can be applied to a larger family of hypergraphs that contains the family of . The next lemma is a generalization of the dependent random choice lemma from [2] and [7]. In the next lemma, we call a vertex and a -subset of vertices of a -uniform hypergraph neighbors if there is some edge of that contains and all of the vertices of . For each vertex and set of vertices , we define to be the set of -subsets of vertices that are neighbors with , and we define to be the set of -subsets of vertices that are neighbors with every vertex in .

Lemma 4.1.

Let be a -uniform hypergraph with vertices and edges. If there is a positive integer such that , then contains a subset of at least vertices such that every vertices in have at least common neighbors among the -subsets of .

Proof.

Pick a set of -subsets of vertices of , choosing -subsets uniformly at random with repetition. Let , and let be the cardinality of . Then , where the second-to-last inequality used Jensen’s inequality.

Let

be the random variable for the number of subsets

of size with fewer than common neighbors among the -subsets of vertices of

. The probability that an arbitrary

-subset is a subset of is , so .

Thus by linearity of expectation, . Thus there exists a choice of for which the corresponding set of cardinality satisfies , so we can remove vertices from to produce a new subset so that all -subsets of have at least common neighbors among the -subsets of . ∎

We can use Lemma 4.1 to get upper bounds for a more general family of -uniform hypergraphs that contains the family of . The next theorem describes one such family.

Theorem 4.2.

For any -uniform hypergraph , let be the -uniform hypergraph obtained by starting with vertices , making disjoint copies of for each -subset of vertices of , and replacing each edge in each with edges of the form for each . For any -uniform hypergraph with and any integers and , we have .

Note that , and that the letter method also works to show that for any integers and . It would be interesting to see if the letter method is useful for other Turán-type problems, and what else can be said about in general.

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