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Weak quasirandomness for uniform hypergraphs
, 2009
"... We study quasirandom properties of kuniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known ChungGrahamWilson theorem fo ..."
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We study quasirandom properties of kuniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known ChungGrahamWilson theorem for quasirandom graphs. Moreover, let Kk be the complete graph on k vertices and M(k) the line graph of the graph of the kdimensional hypercube. We will show that the pair of graphs (Kk, M(k)) has the property that if the number of copies of both Kk and M(k) in another graph G are as expected in the random graph of density d, then G is quasirandom (in the sense of the ChungGrahamWilson theorem) with density close to d.
The QuasiRandomness of Hypergraph Cut Properties
"... Let α1,..., αk satisfy ∑ i αi = 1 and suppose a kuniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets A1,..., Ak of sizes α1n,..., αkn, the number of edges intersecting A1,..., Ak is (asymptotically) the number one would expect to find in a ..."
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Cited by 5 (0 self)
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Let α1,..., αk satisfy ∑ i αi = 1 and suppose a kuniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets A1,..., Ak of sizes α1n,..., αkn, the number of edges intersecting A1,..., Ak is (asymptotically) the number one would expect to find in a random kuniform hypergraph. Can we then infer that H is quasirandom? We show that the answer is negative if and only if α1 = · · · = αk = 1/k. This resolves an open problem raised in 1991 by Chung and Graham [J. AMS ’91]. While hypergraphs satisfying the property corresponding to α1 = · · · = αk = 1/k are not necessarily quasirandom, we manage to find a characterization of the hypergraphs satisfying this property. Somewhat surprisingly, it turns out that (essentially) there is a unique non quasirandom hypergraph satisfying this property. The proofs combine probabilistic and algebraic arguments with results from the theory of association schemes. 1
AN ALGORITHMIC HYPERGRAPH REGULARITY LEMMA
"... Abstract. Szemerédi’s Regularity Lemma is a powerful tools 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 ..."
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Abstract. Szemerédi’s Regularity Lemma is a powerful tools 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 3uniform hypergraphs, which was later extended to kuniform 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. In this paper, we give a constructive proof of the Regularity Lemma for hypergraphs. 1.