Brendan Nagle

Weak hypergraph regularity and linear hypergraphs

We consider conditions which allow the embedding of linear hypergraphs of fixed size. In particular, we prove that any k-uniform hypergraph H of positive uniform density contains all linear k-uniform hypergraphs of a given size. More precisely, we show that for all integers @?>=k>=2 and every d>0 there exists @r>0 for which the following holds: if H is a sufficiently large k-uniform hypergraph with the property that the density of H induced on every vertex subset of size @rn is at least d, then H contains every linear k-uniform hypergraph F with @? vertices. The main ingredient in the proof of this result is a counting lemma for linear hypergraphs, which establishes that the straightforward extension of graph @e-regularity to hypergraphs suffices for counting linear hypergraphs. We also consider some related problems.

Regularity properties for triple systems

Szemeredis Regularity Lemma proved to be a powerful tool in the area of extremal graph theory [J. Komlos and M. Simonovits, Szemeredis 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) = ∪ki=1 Vi, |Vi| = n for all i ∈ [k], and all pairs {Vi, Vj}, 1 ≤ i < j ≤ k, are e-regular of density d, then G contains d??nk(1 + f(e)) cliques K(2)k, where f(e) → 0 as e → 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. Rodl [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 Szemeredis 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.

The asymptotic number of triple systems not containing a fixed one

Abstract For a fixed 3-uniform hypergraph F , call a hypergraph F -free if it contains no subhypergraph isomorphic to F . Let ex (n, F ) denote the size of a largest F -free hypergraph G ⊆[n] 3 . Let F n ( F ) denote the number of distinct labelled F -free G ⊆[n] 3 . We show that F n ( F )=2 ex (n, F )+ o (n 3 ) , and discuss related problems.

Extremal Hypergraph Problems and the Regularity Method

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.

On the Ramsey Number of Sparse 3-Graphs

We consider a hypergraph generalization of a conjecture of Burr and Erdős concerning the Ramsey number of graphs with bounded degree. It was shown by Chvátal, Rödl, Trotter, and Szemerédi [The Ramsey number of a graph with bounded maximum degree, J. Combin. Theory Ser. B 34 (1983), no. 3, 239–243] that the Ramsey number R(G) of a graph G of bounded maximum degree is linear in |V(G)|. We derive the analogous result for 3-uniform hypergraphs.

Integer and fractional packings in dense 3-uniform hypergraphs

Let I0 be any fixed 3-uniform hypergraph. For a 3-uniform hypergraph H we define vI0 (H) to be the maximum size of a set of pairwise triple-disjoint copies of I0 in H. We say a function ψ from the set of copies of I0 in H to [0, 1] is a fractional I0-packing of H if ΣI ∋ e ψ(I) ≤ 1 for every triple e of H. Then v* I 0 (H) is defined to be the maximum value of Σ I ∈ (H I o) ψ(I) over all fractionalI0-packings ψ of H. We show that vI0* (H) - v I0 (H) = o(|V(H)| 3) for all 3-uniform hypergraphs H. This extends the analogous result for graphs, proved by Haxell and Rodl (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 Rodl (2002)], then almost every triple is in about the same number of copies of Kk(3) (the complete 3-uniform hypergraph with k vertices).

Tiling 3‐Uniform Hypergraphs With K43−2e

Let denote the hypergraph consisting of two triples on four points. For an integer n, let denote the smallest integer d so that every 3-uniform hypergraph G of order n with minimum pair-degree contains vertex-disjoint copies of . Kuhn and Osthus (J Combin Theory, Ser B 96(6) (2006), 767–821) proved that holds for large integers n. Here, we prove the exact counterpart, that for all sufficiently large integers n divisible by 4, A main ingredient in our proof is the recent “absorption technique” of Rodl, Rucinski, and Szemeredi (J. Combin. Theory Ser. A 116(3) (2009), 613–636).

An Algorithmic Version of the Hypergraph Regularity Method

Extending the Szemeredi regularity lemma for graphs, P. Frankl and V. Rodl [Random Structures Algorithms, 20 (2002), pp. 131-164] established a 3-graph regularity lemma triple systems

Bounding the strong chromatic index of dense random graphs

{\cal G}_n

On random sampling in uniform hypergraphs

admit bounded partitions of their edge sets, most classes of which consist of regularly distributed triples. Many applications of this lemma require a companion counting lemma [B. Nagle and V. Rodl, Random Structures Algorithms, 23 (2003), pp. 264-332] allowing one to find and enumerate subhypergraphs of a given isomorphism type in a “dense and regular” environment created by the 3-graph regularity lemma. Combined applications of these lemmas are known as the 3-graph regularity method. In this paper, we provide an algorithmic version of the 3-graph regularity lemma which, as we show, is compatible with a counting lemma. We also discuss some applications.