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A simple regularization of hypergraphs
"... Abstract. We give a simple and natural construction of hypergraph regularization. It yields a short proof of a hypergraph regularity lemma. Consequently, as an example of its applications, we have a short selfcontained proof of Szemerédi’s classic theorem on arithmetic progressions (1975) as well a ..."
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Abstract. We give a simple and natural construction of hypergraph regularization. It yields a short proof of a hypergraph regularity lemma. Consequently, as an example of its applications, we have a short selfcontained proof of Szemerédi’s classic theorem on arithmetic progressions (1975) as well as its multidimensional extension by FurstenbergKatznelson (1978). 1.
A SIMPLE REGULARIZATION OF GRAPHS
, 904
"... Abstract. The wellknown regularity lemma of E. Szemerédi for graphs (i.e. 2uniform hypergraphs) claims that for any graph there exists a vertex partition with the property of quasirandomness. We give a simple construction of such a partition. It is done just by taking a constantbounded number of ..."
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Abstract. The wellknown regularity lemma of E. Szemerédi for graphs (i.e. 2uniform hypergraphs) claims that for any graph there exists a vertex partition with the property of quasirandomness. We give a simple construction of such a partition. It is done just by taking a constantbounded number of random vertex samplings only one time (thus, iterationfree). Since it is independent from the definition of quasirandomness, it can be generalized very naturally to hypergraph regularization. In this expository note, we show only a graph case of the paper [5] on hypergraphs, but may help the reader to access [5]. 1.
Linear Ramsey Numbers for boundeddegree hypergraphs
, 2007
"... We show that the the Ramsey number of every boundeddegree uniform hypergraph is linear with respect to the number of vertices. This is a hypergraph extension of the famous theorem for ordinary graphs which Chvátal et al. [8] showed in 1983. Our result may demonstrate the potential of a new hypergr ..."
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We show that the the Ramsey number of every boundeddegree uniform hypergraph is linear with respect to the number of vertices. This is a hypergraph extension of the famous theorem for ordinary graphs which Chvátal et al. [8] showed in 1983. Our result may demonstrate the potential of a new hypergraph regularity lemma by [18].