50 research outputs found

    Counting in hypergraphs via regularity inheritance

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    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

    Extremal results in sparse pseudorandom graphs

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    Szemer\'edi's regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and R\"odl proved an analogue of Szemer\'edi's regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemer\'edi's regularity lemma use an associated counting lemma. In order to prove extensions of these results which also apply to sparse graphs, it remained a well-known open problem to prove a counting lemma in sparse graphs. The main advance of this paper lies in a new counting lemma, proved following the functional approach of Gowers, which complements the sparse regularity lemma of Kohayakawa and R\"odl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse extensions of several well-known combinatorial theorems, including the removal lemmas for graphs and groups, the Erd\H{o}s-Stone-Simonovits theorem and Ramsey's theorem. These results extend and improve upon a substantial body of previous work.Comment: 70 pages, accepted for publication in Adv. Mat

    A sharp threshold for random graphs with a monochromatic triangle in every edge coloring

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    Let R\R be the set of all finite graphs GG with the Ramsey property that every coloring of the edges of GG by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property. Let G(n,p)G(n,p) be the random graph on nn vertices with edge probability pp. We prove that there exists a function c^=c^(n)\hat c=\hat c(n) with 000 0, as nn tends to infinity Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is of independent interest is a generalization of Szemer\'edi's Regularity Lemma to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.

    Extremal and probabilistic results for regular graphs

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    In this thesis we explore extremal graph theory, focusing on new methods which apply to different notions of regular graph. The first notion is dregularity, which means each vertex of a graph is contained in exactly d edges, and the second notion is Szemerédi regularity, which is a strong, approximate version of this property that relates to pseudorandomness. We begin with a novel method for optimising observables of Gibbs distributions in sparse graphs. The simplest application of the method is to the hard-core model, concerning independent sets in d-regular graphs, where we prove a tight upper bound on an observable known as the occupancy fraction. We also cover applications to matchings and colourings, in each case proving a tight bound on an observable of a Gibbs distribution and deriving an extremal result on the number of a relevant combinatorial structure in regular graphs. The results relate to a wide range of topics including statistical physics and Ramsey theory. We then turn to a form of Szemerédi regularity in sparse hypergraphs, and develop a method for embedding complexes that generalises a widely-applied method for counting in pseudorandom graphs. We prove an inheritance lemma which shows that the neighbourhood of a sparse, regular subgraph of a highly pseudorandom hypergraph typically inherits regularity in a natural way. This shows that we may embed complexes into suitable regular hypergraphs vertex-by-vertex, in much the same way as one can prove a counting lemma for regular graphs. Finally, we consider the multicolour Ramsey number of paths and even cycles. A well-known density argument shows that when the edges of a complete graph on kn vertices are coloured with k colours, one can find a monochromatic path on n vertices. We give an improvement to this bound by exploiting the structure of the densest colour, and use the regularity method to extend the result to even cycles

    Transversals via regularity

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    Given graphs G1,…,GsG_1,\ldots,G_s all on the same vertex set and a graph HH with e(H)≤se(H) \leq s, a copy of HH is transversal or rainbow if it contains at most one edge from each GcG_c. When s=e(H)s=e(H), such a copy contains exactly one edge from each GiG_i. We study the case when HH is spanning and explore how the regularity blow-up method, that has been so successful in the uncoloured setting, can be used to find transversals. We provide the analogues of the tools required to apply this method in the transversal setting. Our main result is a blow-up lemma for transversals that applies to separable bounded degree graphs HH. Our proofs use weak regularity in the 33-uniform hypergraph whose edges are those xycxyc where xyxy is an edge in the graph GcG_c. We apply our lemma to give a large class of spanning 33-uniform linear hypergraphs HH such that any sufficiently large uniformly dense nn-vertex 33-uniform hypergraph with minimum vertex degree Ω(n2)\Omega(n^2) contains HH as a subhypergraph. This extends work of Lenz, Mubayi and Mycroft

    A Geometric Theory for Hypergraph Matching

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    We develop a theory for the existence of perfect matchings in hypergraphs under quite general conditions. Informally speaking, the obstructions to perfect matchings are geometric, and are of two distinct types: 'space barriers' from convex geometry, and 'divisibility barriers' from arithmetic lattice-based constructions. To formulate precise results, we introduce the setting of simplicial complexes with minimum degree sequences, which is a generalisation of the usual minimum degree condition. We determine the essentially best possible minimum degree sequence for finding an almost perfect matching. Furthermore, our main result establishes the stability property: under the same degree assumption, if there is no perfect matching then there must be a space or divisibility barrier. This allows the use of the stability method in proving exact results. Besides recovering previous results, we apply our theory to the solution of two open problems on hypergraph packings: the minimum degree threshold for packing tetrahedra in 3-graphs, and Fischer's conjecture on a multipartite form of the Hajnal-Szemer\'edi Theorem. Here we prove the exact result for tetrahedra and the asymptotic result for Fischer's conjecture; since the exact result for the latter is technical we defer it to a subsequent paper.Comment: Accepted for publication in Memoirs of the American Mathematical Society. 101 pages. v2: minor changes including some additional diagrams and passages of expository tex

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    Chromatic thresholds in dense random graphs

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    The chromatic threshold δχ(H,p)\delta_\chi(H,p) of a graph HH with respect to the random graph G(n,p)G(n,p) is the infimum over d>0d > 0 such that the following holds with high probability: the family of HH-free graphs G⊂G(n,p)G \subset G(n,p) with minimum degree δ(G)≥dpn\delta(G) \ge dpn has bounded chromatic number. The study of the parameter δχ(H):=δχ(H,1)\delta_\chi(H) := \delta_\chi(H,1) was initiated in 1973 by Erd\H{o}s and Simonovits, and was recently determined for all graphs HH. In this paper we show that δχ(H,p)=δχ(H)\delta_\chi(H,p) = \delta_\chi(H) for all fixed p∈(0,1)p \in (0,1), but that typically δχ(H,p)≠δχ(H)\delta_\chi(H,p) \ne \delta_\chi(H) if p=o(1)p = o(1). We also make significant progress towards determining δχ(H,p)\delta_\chi(H,p) for all graphs HH in the range p=n−o(1)p = n^{-o(1)}. In sparser random graphs the problem is somewhat more complicated, and is studied in a separate paper.Comment: 36 pages (including appendix), 1 figure; the appendix is copied with minor modifications from arXiv:1108.1746 for a self-contained proof of a technical lemma; accepted to Random Structures and Algorithm

    Local resilience for squares of almost spanning cycles in sparse random graphs

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    In 1962, P\'osa conjectured that a graph G=(V,E)G=(V, E) contains a square of a Hamiltonian cycle if δ(G)≥2n/3\delta(G)\ge 2n/3. Only more than thirty years later Koml\'os, S\'ark\H{o}zy, and Szemer\'edi proved this conjecture using the so-called Blow-Up Lemma. Here we extend their result to a random graph setting. We show that for every ϵ>0\epsilon > 0 and p=n−1/2+ϵp=n^{-1/2+\epsilon} a.a.s. every subgraph of Gn,pG_{n,p} with minimum degree at least (2/3+ϵ)np(2/3+\epsilon)np contains the square of a cycle on (1−o(1))n(1-o(1))n vertices. This is almost best possible in three ways: (1) for p≪n−1/2p\ll n^{-1/2} the random graph will not contain any square of a long cycle (2) one cannot hope for a resilience version for the square of a spanning cycle (as deleting all edges in the neighborhood of single vertex destroys this property) and (3) for c<2/3c<2/3 a.a.s. Gn,pG_{n,p} contains a subgraph with minimum degree at least cnpcnp which does not contain the square of a path on (1/3+c)n(1/3+c)n vertices
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