Regularity Lemma For Hypergraphs Research Articles

Hypergraph regularity and random sampling

AbstractSuppose that a ‐uniform hypergraph satisfies a certain regularity instance (that is, there is a partition of given by the hypergraph regularity lemma into a bounded number of quasirandom subhypergraphs of prescribed densities). We prove that with high probability a large enough uniform random sample of the vertex set of also admits the same regularity instance. Here the crucial feature is that the error term measuring the quasirandomness of the subhypergraphs requires only an arbitrarily small additive correction. This has applications to combinatorial property testing. The graph case of the sampling result was proved by Alon, Fischer, Newman and Shapira.

Weak hypergraph regularity and applications to geometric Ramsey theory

Let Δ = Δ 1 × … × Δ d ⊆ R n \Delta =\Delta _1\times \ldots \times \Delta _d\subseteq \mathbb {R}^n , where R n = R n 1 × ⋯ × R n d \mathbb {R}^n=\mathbb {R}^{n_1}\times \cdots \times \mathbb {R}^{n_d} with each Δ i ⊆ R n i \Delta _i\subseteq \mathbb {R}^{n_i} a non-degenerate simplex of n i n_i points. We prove that any set S ⊆ R n S\subseteq \mathbb {R}^n , with n = n 1 + ⋯ + n d n=n_1+\cdots +n_d of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of the configuration Δ \Delta . In particular any such set S ⊆ R 2 d S\subseteq \mathbb {R}^{2d} contains a d d -dimensional cube of side length λ \lambda , for all λ ≥ λ 0 ( S ) \lambda \geq \lambda _0(S) . We also prove analogous results with the underlying space being the integer lattice. The proof is based on a weak hypergraph regularity lemma and an associated counting lemma developed in the context of Euclidean spaces and the integer lattice.

A Tight Bound for Hyperaph Regularity

The hypergraph regularity lemma—the extension of Szemeredi’s graph regularity lemma to the setting of k-uniform hypergraphs—is one of the most celebrated combinatorial results obtained in the past decade. By now there are several (very different) proofs of this lemma, obtained by Gowers, by Nagle–Rodl–Schacht–Skokan and by Tao. Unfortunately, what all these proofs have in common is that they yield regular partitions whose order is given by the k-th Ackermann function. We show that such Ackermann-type bounds are unavoidable for every $$k \ge 2$$ , thus confirming a prediction of Tao. Prior to our work, the only result of this type was Gowers’ famous lower bound for graph regularity.

A sparse regular approximation lemma

We introduce a new variant of Szemeredi's regularity which we call the sparse regular approximation lemma (SRAL). The input to this is a graph $G$ of edge density $p$ and parameters $\epsilon, \delta$, where we think of $\delta$ as a constant. The goal is to construct an $\epsilon$-regular partition of $G$ while having the freedom to add/remove up to $\delta |E(G)|$ edges. As we show here, this weaker variant of the regularity already suffices for proving the graph removal and the hypergraph regularity lemma, which are two of the main applications of the (standard) regularity lemma. This of course raises the following question: can one obtain quantitative bounds for SRAL that are significantly better than those associated with the regularity lemma? Our first result answers the above question affirmatively by proving an upper bound for SRAL given by a tower of height $O(\log 1/p)$. This allows us to reprove Fox's upper bound for the graph removal lemma. Our second result is a matching lower bound for SRAL showing that a tower of height $\Omega(\log 1/p)$ is unavoidable. We in fact prove a more general multicolored lower bound which is essential for proving lower bounds for the hypergraph regularity lemma.

An algorithmic hypergraph regularity lemma

AbstractSzemerédi 's Regularity Lemma is a powerful tool in graph theory. It asserts that all large graphs admit bounded partitions of their edge sets, most classes of which consist of uniformly distributed edges. The original proof of this result was nonconstructive, and a constructive proof was later given by Alon, Duke, Lefmann, Rödl, and Yuster. Szemerédi's Regularity Lemma was extended to hypergraphs by various authors. Frankl and Rödl gave one such extension in the case of 3‐uniform hypergraphs, which was later extended to k‐uniform hypergraphs by Rödl and Skokan. W.T. Gowers gave another such extension, using a different concept of regularity than that of Frankl, Rödl, and Skokan. Here, we give a constructive proof of a regularity lemma for hypergraphs.

The minimum vertex degree for an almost-spanning tight cycle in a 3-uniform hypergraph

We prove that any 3-uniform hypergraph whose minimum vertex degree is at least 59+o(1)n2 admits an almost-spanning tight cycle, that is, a tight cycle leaving o(n) vertices uncovered. The bound on the vertex degree is asymptotically best possible. Our proof uses the hypergraph regularity method, and in particular a recent version of the hypergraph regularity lemma proved by Allen, Böttcher, Cooley and Mycroft.

Hypergraph Independent Sets

The study of extremal problems related to independent sets in hypergraphs is a problem that has generated much interest. There are a variety of types of independent sets in hypergraphs depending on the number of vertices from an independent set allowed in an edge. We say that a subset of vertices isj-independentif its intersection with any edge has size strictly less thanj. The Kruskal–Katona theorem implies that in anr-uniform hypergraph with a fixed size and order, the hypergraph with the mostr-independent sets is the lexicographic hypergraph. In this paper, we use a hypergraph regularity lemma, along with a technique developed by Loh, Pikhurko and Sudakov, to give an asymptotically best possible upper bound on the number ofj-independent sets in anr-uniform hypergraph.

Almost all triangle-free triple systems are tripartite

A triangle in a triple system is a collection of three edges isomorphic to {123,124,345}. A triple system is triangle-free if it contains no three edges forming a triangle. It is tripartite if it has a vertex partition into three parts such that every edge has exactly one point in each part. It is easy to see that every tripartite triple system is triangle-free. We prove that almost all triangle-free triple systems with vertex set [n] are tripartite. Our proof uses the hypergraph regularity lemma of Frankl and Rodl [13], and a stability theorem for triangle-free triple systems due to Keevash and the second author [15].

Almost all triple systems with independent neighborhoods are semi-bipartite

The neighborhood of a pair of vertices u, v in a triple system is the set of vertices w such that uvw is an edge. A triple system H is semi-bipartite if its vertex set contains a vertex subset X such that every edge of H intersects X in exactly two points. It is easy to see that if H is semi-bipartite, then the neighborhood of every pair of vertices in H is an independent set. We show a partial converse of this statement by proving that almost all triple systems with vertex sets [ n ] and independent neighborhoods are semi-bipartite. Our result can be viewed as an extension of the Erdős–Kleitman–Rothschild theorem to triple systems. The proof uses the Frankl–Rödl hypergraph regularity lemma, and stability theorems. Similar results have recently been proved for hypergraphs with various other local constraints.

Dirac-type results for loose Hamilton cycles in uniform hypergraphs

A classic result of G.A. Dirac in graph theory asserts that for n ⩾ 3 every n-vertex graph with minimum degree at least n / 2 contains a spanning (so-called Hamilton) cycle. G.Y. Katona and H.A. Kierstead suggested a possible extension of this result for k-uniform hypergraphs. There a Hamilton cycle of an n-vertex hypergraph corresponds to an ordering of the vertices such that every k consecutive (modulo n) vertices in the ordering form an edge. Moreover, the minimum degree is the minimum ( k − 1 ) -degree, i.e. the minimum number of edges containing a fixed set of k − 1 vertices. V. Rödl, A. Ruciński, and E. Szemerédi verified (approximately) the conjecture of Katona and Kierstead and showed that every n-vertex, k-uniform hypergraph with minimum ( k − 1 ) -degree ( 1 / 2 + o ( 1 ) ) n contains such a tight Hamilton cycle. We study the similar question for Hamilton ℓ-cycles. A Hamilton ℓ-cycle in an n-vertex, k-uniform hypergraph ( 1 ⩽ ℓ < k ) is an ordering of the vertices and an ordered subset of the edges such that each such edge corresponds to k consecutive (modulo n) vertices and two consecutive edges intersect in precisely ℓ vertices. We prove sufficient minimum ( k − 1 ) -degree conditions for Hamilton ℓ-cycles if ℓ < k / 2 . In particular, we show that for every ℓ < k / 2 every n-vertex, k-uniform hypergraph with minimum ( k − 1 ) -degree ( 1 / ( 2 ( k − ℓ ) ) + o ( 1 ) ) n contains such a loose Hamilton ℓ-cycle. This degree condition is approximately tight and was conjectured by D. Kühn and D. Osthus (for ℓ = 1 ), who verified it when k = 3 . Our proof is based on the so-called weak regularity lemma for hypergraphs and follows the approach of V. Rödl, A. Ruciński, and E. Szemerédi.

The Ramsey Number for 3-Uniform Tight Hypergraph Cycles

LetC(3)ndenote the 3-uniformtight cycle, that is, the hypergraph with verticesv1, .–.–.,vnand edgesv1v2v3,v2v3v4, .–.–.,vn−1vnv1,vnv1v2. We prove that the smallest integerN=N(n) for which every red–blue colouring of the edges of the complete 3-uniform hypergraph withNvertices contains a monochromatic copy ofC(3)nis asymptotically equal to 4n/3 ifnis divisible by 3, and 2notherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl.

A correspondence principle between (hyper)graph theory and probability theory, and the (hyper)graph removal lemma

We introduce a correspondence principle (analogous to the Furstenberg Correspondence Principle) that allows one to extract an infinite random graph or hypergraph from a sequence of increasingly large deterministic graphs or hypergraphs. As an application we present a new (infinitary) proof of the hypergraph removal lemma of Nagle-Schacht-Rodl-Skokan and Gowers, which does not require the hypergraph regularity lemma and requires significantly less computation. This in turn gives new proofs of several corollaries of the hypergraph removal lemma, such as Szemeredi’s Theorem on arithmetic progressions.

Regular Partitions of Hypergraphs: Regularity Lemmas

Szemerédi's regularity lemma for graphs has proved to be a powerful tool with many subsequent applications. The objective of this paper is to extend the techniques developed by Nagle, Skokan, and the authors and obtain a stronger and more ‘user-friendly’ regularity lemma for hypergraphs.

A Quantitative Ergodic Theory Proof of Szemerédi's Theorem

A famous theorem of Szemerédi asserts that given any density $0 &lt; \delta \leq 1$ and any integer $k \geq 3$, any set of integers with density $\delta$ will contain infinitely many proper arithmetic progressions of length $k$. For general $k$ there are essentially four known proofs of this fact; Szemerédi's original combinatorial proof using the Szemerédi regularity lemma and van der Waerden's theorem, Furstenberg's proof using ergodic theory, Gowers' proof using Fourier analysis and the inverse theory of additive combinatorics, and the more recent proofs of Gowers and Rödl-Skokan using a hypergraph regularity lemma. Of these four, the ergodic theory proof is arguably the shortest, but also the least elementary, requiring passage (via the Furstenberg correspondence principle) to an infinitary measure preserving system, and then decomposing a general ergodic system relative to a tower of compact extensions. Here we present a quantitative, self-contained version of this ergodic theory proof, and which is "elementary" in the sense that it does not require the axiom of choice, the use of infinite sets or measures, or the use of the Fourier transform or inverse theorems from additive combinatorics. It also gives explicit (but extremely poor) quantitative bounds.

Szemer'edi's regularity lemma revisited

Szemer'edi's regularity lemma is a basic tool in graph theory, and also plays an important role in additive combinatorics, most notably in proving Szemer\'edi's theorem on arithmetic progressions . In this note we revisit this lemma from the perspective of probability theory and information theory instead of graph theory, and observe a variant of this lemma which introduces a new parameter $F$. This stronger version of the regularity lemma was iterated in a recent paper of the author to reprove the analogous regularity lemma for hypergraphs.

Regularity, uniformity, and quasirandomness

Rodl et al. extend a powerful tool, the regularity lemma, from graphs to hypergraphs. Graph theory is the appropriate language for discussing binary relations on objects. Results in graph theory have numerous applications in biology, chemistry, computer science, and physics. In cases of multiple relations, instead of binary relations more general structures known as hypergraphs are the right tools. However, it turns out that because of their extremely complex structure, hypergraphs are very difficult to deal with. As with number theory, there are questions about hypergraphs that are easy to state but very difficult to answer. In this issue of PNAS, Rodl et al. (1) extend a powerful tool, the regularity lemma, from graphs to hypergraphs. Contrary to the general terminology, in extremal graph theory regularity is a measure of randomness. Random graphs are easy to work with, especially when one wants to estimate the (expected) number of small subgraphs. In complex structures, like in dense graphs, one can substitute randomness with weaker but still useful properties. The motivation behind graph regularity is to arrange the vertices of a graph in such a way that the graph becomes similar to the union of a few random graphs, and then one can apply standard counting methods from probability theory. In order to define hypergraph regularity, one has to introduce somehow complicated and technical notations. However, even without these notations we can formulate the most important consequence of the so-called hypergraph regularity method. The method, which is the combination of the hypergraph regularity lemma and a counting lemma is described by Rodl et al. (1). Similar results with the same consequences have been obtained independently by Gowers (2). Inspired by the methods of refs. 1 and 2, very recently Tao (T. Tao, personal communication) gave another proof of the main results. …

The hypergraph regularity method and its applications

Szemeredi's regularity lemma asserts that every graph can be decomposed into relatively few random-like subgraphs. This random-like behavior enables one to find and enumerate subgraphs of a given isomorphism type, yielding the so-called counting lemma for graphs. The combined application of these two lemmas is known as the regularity method for graphs and has proved useful in graph theory, combinatorial geometry, combinatorial number theory, and theoretical computer science. Here, we report on recent advances in the regularity method for k-uniform hypergraphs, for arbitrary k > or = 2. This method, purely combinatorial in nature, gives alternative proofs of density theorems originally due to E. Szemeredi, H. Furstenberg, and Y. Katznelson. Further results in extremal combinatorics also have been obtained with this approach. The two main components of the regularity method for k-uniform hypergraphs, the regularity lemma and the counting lemma, have been obtained recently: Rodl and Skokan (based on earlier work of Frankl and Rodl) generalized Szemeredi's regularity lemma to k-uniform hypergraphs, and Nagle, Rodl, and Schacht succeeded in proving a counting lemma accompanying the Rodl-Skokan hypergraph regularity lemma. The counting lemma is proved by reducing the counting problem to a simpler one previously investigated by Kohayakawa, Rodl, and Skokan. Similar results were obtained independently by W. T. Gowers, following a different approach.

Regularity Lemma for k‐uniform hypergraphs

AbstractSzemerédi's Regularity Lemma proved to be a very powerful tool in extremal graph theory with a large number of applications. Chung [Regularity lemmas for hypergraphs and quasi‐randomness, Random Structures Algorithms 2 (1991), 241–252], Frankl and Rödl [The uniformity lemma for hypergraphs, Graphs Combin 8 (1992), 309–312; Extremal problems on set systems, Random Structures Algorithms 20 (2002), 131–164] considered several extensions of Szemerédi's Regularity Lemma to hypergraphs. In particular, [Extremal problems on set systems, Random Structures Algorithms 20 (2002), 131–164] contains a regularity lemma for 3‐uniform hypergraphs that was applied to a number of problems. In this paper, we present a generalization of this regularity lemma to k‐uniform hypergraphs. Similar results were recently independently and alternatively obtained by W. T. Gowers. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004

Regularity properties for triple systems

AbstractSzemerédi's Regularity Lemma proved to be a powerful tool in the area of extremal graph theory J. Komlós and M. Simonovits, Szemerédi's Regularity Lemma and its applications in graph theory, Combinatorics 2 (1996), 295–352. Many of its applications are based on the following technical fact: If G is a k‐partite graph with V(G) = ∪Vi, |Vi| = n for all i ∈ [k], and all pairs {Vi, Vj}, 1 ≤ i &lt; j ≤ k, are ϵ‐regular of density d, then G contains dnk(1 + f(ϵ)) cliques K, where f(ϵ) → 0 as ϵ → 0. The aim of this paper is to establish the analogous statement for 3‐uniform hypergraphs. Our result, to which we refer as The Counting Lemma, together with Theorem 3.5 of P. Frankl and V. Rödl [Extremal problems on set systems, Random Structures Algorithms 20(2) (2002), 131–164, a Regularity Lemma for Hypergraphs, can be applied in various situations as Szemerédi's Regularity Lemma is for graphs. Some of these applications are discussed in previous papers, as well as in upcoming papers, of the authors and others. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 264–332, 2003

Integer and fractional packings in dense 3‐uniform hypergraphs

AbstractLet 𝒥0 be any fixed 3‐uniform hypergraph. For a 3‐uniform hypergraph ℋ︁ we define ν(ℋ︁) to be the maximum size of a set of pairwise triple‐disjoint copies of 𝒥0 in ℋ︁. We say a function ψ from the set of copies of 𝒥0 in ℋ︁ to [0, 1] is a fractional 𝒥0‐packing of ℋ︁ if ∑𝒥∋e ψ(𝒥) ≤ 1 for every triple e of ℋ︁. Then ν(ℋ︁) is defined to be the maximum value of ∑ ψ(𝒥) over all fractional 𝒥0‐packings ψ of ℋ︁. We show that ν(ℋ︁) − ν(ℋ︁) = o(|V(ℋ︁)| 3) for all 3‐uniform hypergraphs ℋ︁. This extends the analogous result for graphs, proved by Haxell and Rödl (2001), and requires a significant amount of new theory about regularity of 3‐uniform hypergraphs. In particular, we prove a result that we call the Extension Theorem. This states that if a k‐partite 3‐uniform hypergraph is regular [in the sense of the hypergraph regularity lemma of Frankl and Rödl (2002)], then almost every triple is in about the same number of copies of K (the complete 3‐uniform hypergraph with k vertices). © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 22: 248–310, 2003