We present a deterministic algorithm A that, in O(m 2) time, verifies whether a given m by m bipartite graph G is regular, in the sense of Szemerédi [E. Szemerédi, Regular partitions of graphs, Problèmes Combinatoires et Théorie des Graphes (Colloq. Internat. CNRS, Univ. Orsay,

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 ℓ ≥ k ≥ 2 and every d> 0 there exists ϱ> 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 ϱn is at least d, then H contains every linear k-uniform hypergraph F with ℓ 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 ε-regularity to hypergraphs suffices for counting linear hypergraphs. We also consider some related problems.

Let R be the set of all finite graphs G with the Ramsey property that every coloring of the edges of G 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) be the random graph on n vertices with edge probability p. We prove that there exists a function �c = �c(n) with 0 <c<�c<Csuch that for any ε>0, as n tends to infinity and Pr � G(n, (1 − ε)�c / √ n) ∈R � →0 Pr � G(n, (1 + ε)�c / √ n) ∈R � →1. A crucial tool that is used in the proof and is of independent interest is a generalization of Szemerédi’s Regularity Lemma to a certain hypergraph setting.

Recently, Bollobás, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with Θ(n) edges. Roughly speaking, there is one model for each kernel, i.e., each symmetric measurable function from [0,1] 2 to the non-negative reals, although the details are much more complicated, to ensure the exact inclusion of many of the recent models for large-scale real-world networks. A different connection between kernels and random graphs arises in the recent work of Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi. They introduced several natural metrics on dense graphs (graphs with n vertices and Θ(n 2) edges), showed that these metrics are equivalent, and gave a description of the completion of the space of all graphs with respect to any of these metrics in terms of graphons, which are essentially kernels. One of the most appealing aspects of this work is the message that sequences of inhomogeneous quasi-random graphs are in a sense completely

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In this paper we determine the local and global resilience of random graphs Gn,p (p ≫ n −1) with respect to the property of containing a cycle of length at least (1 − α)n. Roughly speaking, given α> 0, we determine the smallest rg(G, α) with the property that almost surely every subgraph of G = Gn,p having more than rg(G, α)|E(G) | edges contains a cycle of length at least (1 − α)n (global resilience). We also obtain, for α < 1/2, the smallest rl(G, α) such that any H ⊆ G having deg H(v) larger than rl(G, α) deg G(v) for all v ∈ V (G) contains a cycle of length at least (1 − α)n (local resilience). The results above are in fact proved in the more general setting of pseudorandom graphs. Supported by a CAPES–Fulbright scholarship. Partially supported by FAPESP and CNPq through a Temático–ProNEx project

Abstract. Szemerédi’s regularity lemma proved to be a fundamental result in modern graph theory. It had a number of important applications and is a widely used tool in extremal combinatorics. For some applications variants of the regularity lemma were considered. Here we discuss several of those variants and their relation to each other. 1.

Abstract. We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity. Contents

We study the following one-person game against a random graph: the Player's goal is to 2-colour a random sequence of edges e1, e2,... of a complete graph on n vertices, avoiding a monochromatic triangle for as long as possible. The game is over when a monochromatic triangle is created. The online version of the game requires that the Player should colour each edge when it comes, before looking at the next edge. While it is not hard to prove that the expected length of this game is about n

a given graph