We develop a theory of regularity inheritance in 3-uniform hypergraphs. As a simple consequence we deduce a strengthening of a counting lemma of Frankl and Rödl. We believe that the approach is sufficiently flexible and general to permit extensions of our results in the direction of a hypergraph blow-up lemma

Davies, Ewan

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  Ewan Davies  Counting in hypergraphs via regularity inheritance  Article (Accepted version) (Refereed)     Original citation: Davies, Ewan (2015) Counting in hypergraphs via regularity inheritance. Electronic Notes in Discrete Mathematics, 49. pp. 413-417. ISSN 1571-0653  DOI: 10.1016/j.endm.2015.06.058     Reuse of this item is permitted through licensing under the Creative Commons:  © 2015 Elsevier CC-BY-NC-ND  This version available at: http://eprints.lse.ac.uk/65062/ Available in LSE Research Online: January 2016  LSE has developed LSE Research Online so that users may access research output of the School. Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. You may freely distribute the URL (http://eprints.lse.ac.uk) of the LSE Research Online website.     Counting in hypergraphs via regularityinheritanceEwan Davies 1Department of MathematicsLondon School of Economics and Political ScienceLondon WC2A 2AE, UKAbstractWe develop a theory of regularity inheritance in 3-uniform hypergraphs. As a simpleconsequence we deduce a strengthening of a counting lemma of Frankl and Ro¨dl.We believe that the approach is sufficiently flexible and general to permit extensionsof our results in the direction of a hypergraph blow-up lemma.Keywords: Hypergraph regularity, counting lemma, regularity inheritance1 IntroductionSzemere´di’s regularity lemma [15] states that any graph G has a ‘regularpartition’ into a bounded number of pieces, almost all of which are quasirandom.An accompanying counting lemma tells us that the number of copies of anysmall subgraph in G is approximately the same as would be expected if thepieces were genuinely random. There are various generalisations to hypergraphs(e.g. [6,7,14] which have been instrumental in solving important problems.1 Email: E.S.Davies@lse.ac.ukThe blow-up lemma of Komlo´s, Sa´rko¨zy and Szemere´di [9] is a powerfuldevelopment of the counting lemma. It uses a randomised embedding algorithmto show that we can actually embed certain spanning subgraphs into suitablepieces of the partition produced by the regularity lemma. A technical conditioncalled ‘image restrictions’ is permitted; a useful feature for applications, e.g. [1].Keevash [8] proved a hypergraph blow-up lemma, but there are significanttechnical difficulties in establishing and applying the result. Complexityarises from the kind of hypergraph regularity used, known as the regularapproximation lemma of Ro¨dl and Schacht [13]. Furthermore, Keevash’s resultonly allows for a weak form of image restriction which makes it hard to use.We describe a simple proof of a counting lemma for 3-uniform hypergraphs(3-graphs) using regularity inheritance, a technique we develop. The merits ofour approach are connected with the simple kind of 3-graph regularity we use.A hypergraph blow-up lemma proved using these ideas is a work in progress.In particular, we seek to allow a number of vertices and pairs to be suitablyimage restricted, stronger than the restriction permitted by Keevash [8].In future work we seek to generalise our techniques to k-uniform hypergraphsfor k > 3. There are significant difficulties here, for some of the tools we useto establish regularity inheritance lemmas are not yet well-developed in higheruniformities. In particular a local characterisation of hypergraph regularity isimportant for our methods, and for algorithmic proofs. We also hope that ourapproach will generalise to the ‘sparse setting’, that is, when we work with ahypergraph which is a relatively dense subgraph of a sparse but pseudorandomhypergraph. A counting lemma for this setting was recently proved by Conlon,Fox and Zhao [3], but no counterpart blow-up lemma is known.2 ResultsOur notation resembles that of [3]. We identify graphs and 3-graphs withtheir edge sets. The neighbourhood of a vertex x in a graph or 3-graph F isF (x) := {e \ x : x ∈ e ∈ F}, and we write F [W ] for the induced subgraphon vertex set W . Let ∂e = {f ( e : |f | = |e| − 1}. Functions gf , he are theindicators for edge sets of graphs G and 3-graphs H respectively. All setsare finite. For an index set J and vertex sets {Vj}j∈J , let VJ := ∏j∈J Vj. Wewrite xJ for an element of VJ , that is, the vector (xj)j∈J with xj ∈ Vj. WriteE[g(xJ)|xJ ∈ VJ ] for the expectation over xJ chosen uniformly at random fromVJ . For statements involving positive real parameters, let a = b ± c meanb− c ≤ a ≤ b+ c, and write a b to mean there is an increasing function fso that the argument is valid for 0 < a < f(b).Definition 2.1 Let f = {1, 2}, and {Vj}j∈f be vertex sets. Let G be abipartite graph with bipartition V1, V2 and indicator function gf : Vf → {0, 1}.We say gf is (ε2, d2)-regular if, for all j ∈ ∂f and functions uj : Vj → [0, 1], wehave ∣∣∣∣E[(gf (xf )− d2)u1(x1)u2(x2)∣∣∣xf ∈ Vf]∣∣∣∣ ≤ ε2.Although this definition looks stronger than the ‘usual’ definition, whereeach uj takes values in {0, 1}, it is easy to prove that they are equivalent.Definition 2.2 Let e = {1, 2, 3} and {Vj}j∈e be vertex sets. Let G be atripartite graph with vertex partition V1, V2, V3, and indicator functionsgf : Vf → {0, 1} which are (ε2, d2)-regular for f ∈ ∂e. Let H be a 3-graphwith indicator function he ≤ ∏f∈∂e gf pointwise, so edges of H are triangles inG. We say he is (ε3, d3)-regular relative to G if, for all f ∈ ∂e and functionsuf : Vf → [0, 1] with uf ≤ gf pointwise, we have∣∣∣∣E[(he(xe)− d3) ∏f∈∂euf (xf )∣∣∣xe ∈ Ve]∣∣∣∣ ≤ ε3d32.We also say H is (ε3, d3)-regular with respect to G when we do not explicitlydefine the indicator function he.Our main result is a strengthening of the 3-graph counting lemma developedpreviously by Ro¨dl and coauthors Frankl, Nagle, Peng and Skokan [6,11,12].These results require a stronger form of regularity than the above Definition 2.2in which the edges of H are approximately uniformly distributed over trianglesin unions of r subgraphs of G, for some r ∈ N which is large compared to1/d2. Taking r = 1 in this formulation results in a strictly weaker regularitywhich is equivalent to our setting. This appeal of working with the weaker3-graph regularity is a local characterisation in terms of subgraph counts. Thischaracterisation does not hold for large values of r in the stronger 3-graphregularity, thus working with r = 1 recovers the useful property; see [4,10].Our approach differs substantially from that of Nagle and Ro¨dl [11] andwe remove the need to apply any regularity lemma within the proof of ourcounting lemma. Another approach, due to Peng, Ro¨dl and Skokan [12] isto transfer to the ‘dense’ or ‘absolute quasirandom’ setting in which d2 = 1and appeal to a dense counting lemma. This idea is relevant to our work, aswe depend on a technical result that is proved by a similar transference [10].Since we work with the weaker regularity of Definition 2.2 our methods arestronger than those of [11,12]. Moreover, our proof resembles an embeddingprocess used in the blow-up lemma, a feature we hope to exploit in due course.Theorem 2.3 (Counting lemma) Let J be a set and F be a 3-graph on J .Write ∂F for the union of ∂e over e ∈ F . Let {Vj}j∈J be vertex sets each of sizeat least n. For constants 1n ε2  d2  ε3  ε′3  d3, the following holds.Let G be a graph with indicators gf : Vf → {0, 1} which are (ε2, d2)-regularfor all f ∈ ∂F . Let H be a 3-graph with indicators he : Ve → {0, 1} which are(ε3, d3)-regular with respect to G for all e ∈ F .ThenE[∏e∈Fhe(xe)∣∣∣x ∈ VJ] = d|F |3 d|∂F |2 ± ε′3d|∂F |2 .To obtain Theorem 2.3 we adapt the following sketch of the counting lemmain dense graphs. For simplicity, consider G from the setup of Theorem 2.3 andlet ∂F be a triangle on vertices {1, 2, 3}. We apply a fact commonly knownas ‘slicing’ or ‘regular restriction’ to G[V2, V3], which is is (ε2, d2)-regular. IfWj ⊂ Vj for j = 2, 3 are subsets of size at least α|Vj|, the induced subgraphG[W2,W3] is (ε2/α2, d2)-regular. If the Wj are G-neighbourhoods of a typicalx1 ∈ V1, we would have α = d2 ± ε2 and so given ε2  d2, we say theneighbourhood inherits regularity. To count triangles, sum over x1 ∈ V1 thenumber of edges in G[G(x1)], which we estimate by the inherited regularity.Applying the regularity lemma of Frankl and Ro¨dl [5], one can only hopeto achieve the relation d2  ε3 between parameters in Definition 2.2. Hence G-neighbourhoods are typically too small for the regularity of H to directly controledges upon them. An analogue of this problem also occurs in sparse graphs,however Conlon, Fox and Zhao [2] proved a form of regularity inheritancefor that setting. We generalise their approach to hypergraphs and prove newinheritance lemmas such as 2.4 below. We deduce Theorem 2.3 from theselemmas by an appropriate generalisation of the above sketch for graphs.To state an inheritance lemma, consider the setup of Theorem 2.3 and let0 ∈ J . We show that almost all x0 ∈ V0 have the property that H[G(x0)]is regular with respect to G[G(x0)]. The parameter measuring the inheritedregularity is ε′3  d3, hence this result is much stronger than what one candeduce directly from Definition 2.2 using the uf to indicate edges of G[G(x0)].Lemma 2.4 (3-sided inheritance) Let J = {0, 1, 2, 3}, e = {1, 2, 3}, andF be the complete graph on J . Let 1n ε2  d2  ε3  ε′3  d3, and{Vj}j∈J be vertex sets each of size at least n. Let G be a graph with indicatorsgf : Vf → {0, 1} which are (ε2, d2)-regular for f ∈ F , and let H be a 3-graphon Ve with indicator he ≤ ∏f∈∂e gf which is (ε3, d3)-regular with respect to G.Then for all but at most ε′3|V0| vertices x0 ∈ V0, the induced 3-graphH[G(x0)] is (ε′3, d3)-regular with respect to G[G(x0)].References[1] Bo¨ttcher, J., M. Schacht and A. Taraz, Proof of the bandwidth conjecture ofBolloba´s and Komlo´s, Math. Ann. 343 (2009), 175–205.[2] Conlon, D., J. Fox and Y. Zhao, Extremal results in sparse pseudorandom graphs,Adv. Math. 256 (2014), 206–290.[3] Conlon, D., J. Fox and Y. Zhao, A relative Szemere´di theorem, Geom. Funct.Anal. (2015), 1–30.[4] Dementieva, Y., P. E. Haxell, B. Nagle and V. Ro¨dl, On characterizing hypergraphregularity, Random Structures & Algorithms 21 (2002), 293–335.[5] Frankl, P. and V. Ro¨dl, The Uniformity Lemma for hypergraphs, Graphs andCombinatorics 8 (1992), 309–312.[6] Frankl, P. and V. Ro¨dl, Extremal problems on set systems, Random Structures& Algorithms 20 (2002), 131–164.[7] Gowers, W. T., Quasirandomness, counting and regularity for 3-uniformhypergraphs, Combin., Probab. Comput. 15 (2006), 143–184.[8] Keevash, P., A hypergraph blow-up lemma, Random Structures & Algorithms39 (2011), 275–376.[9] Komlo´s, J., G. N. Sa´rko¨zy and E. Szemere´di, Blow-up lemma, Combinatorica17 (1997), 109–123.[10] Nagle, B., A. Poerschke, V. Ro¨dl and M. Schacht, Hypergraph regularity andquasi-randomness, in: Proc. of the 20th Annual ACM-SIAM Symp. on DiscreteAlgorithms, 2009, 227–235.[11] Nagle, B. and V. Ro¨dl, Regularity properties for triple systems, RandomStructures & Algorithms 23 (2003), 264–332.[12] Peng, Y., V. Ro¨dl and J. Skokan, Counting small cliques in 3-uniformhypergraphs, Combin., Probab. Comput. 14 (2005), 371–413.[13] Ro¨dl, V. and M. Schacht, Regular partitions of hypergraphs: regularity lemmas,Combin., Probab. Comput. 16 (2007), 833–885.[14] Ro¨dl, V. and J. Skokan, Regularity Lemma for k-uniform hypergraphs, RandomStructures & Algorithms 25 (2004), 1–42.[15] Szemere´di, E., Regular partitions of Graphs, in: Proble`me Combinatoire etThe`orie des Graphes, (Colloq. Internat. CNRS, Univ. Orsay, Orsay 1976) 260,1978, 399–401.

Counting in hypergraphs via regularity inheritance

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