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28
A new proof of the density HalesJewett theorem
, 2009
"... The Hales–Jewett theorem asserts that for every r and every k there exists n such that every rcolouring of the ndimensional grid {1,..., k} n contains a combinatorial line. This result is a generalization of van der Waerden’s theorem, and it is one of the fundamental results of Ramsey theory. The ..."
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Cited by 40 (2 self)
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The Hales–Jewett theorem asserts that for every r and every k there exists n such that every rcolouring of the ndimensional grid {1,..., k} n contains a combinatorial line. This result is a generalization of van der Waerden’s theorem, and it is one of the fundamental results of Ramsey theory. The theorem of van der Waerden has a famous density version, conjectured by Erdős and Turán in 1936, proved by Szemerédi in 1975 and given a different proof by Furstenberg in 1977. The Hales–Jewett theorem has a density version as well, proved by Furstenberg and Katznelson in 1991 by means of a significant extension of the ergodic techniques that had been pioneered by Furstenberg in his proof of Szemerédi’s theorem. In this paper, we give the first elementary proof of the theorem of Furstenberg and Katznelson, and the first to provide a quantitative bound on how large n needs to be. In particular, we show that a subset of [3] n of density δ contains a combinatorial line if n ≥ 2 ⇈ O(1/δ 3). Our proof is surprisingly simple: indeed, it gives what is probably the simplest known proof of Szemerédi’s theorem.
D.: Embeddings and Ramsey numbers of sparse kuniform hypergraphs, Combinatorica 29
, 2009
"... Abstract. Chvátal, Rödl, Szemerédi and Trotter [3] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In [5, 19] the same result was proved for 3uniform hypergraphs. Here we extend this result to kuniform hypergraphs for any integer k ≥ 3. As in the 3uni ..."
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Cited by 19 (4 self)
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Abstract. Chvátal, Rödl, Szemerédi and Trotter [3] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In [5, 19] the same result was proved for 3uniform hypergraphs. Here we extend this result to kuniform hypergraphs for any integer k ≥ 3. As in the 3uniform case, the main new tool which we prove and use is an embedding lemma for kuniform hypergraphs of bounded maximum degree into suitable kuniform ‘quasirandom ’ hypergraphs.
Hamilton ℓcycles in uniform hypergraphs
 JOURNAL OF COMBINATORIAL THEORY. SERIES A
"... We say that a kuniform hypergraph C is an ℓcycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of consecutive edges (in the natural ordering of the edges) intersects in precisely ℓ vertices. We prove th ..."
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Cited by 12 (3 self)
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We say that a kuniform hypergraph C is an ℓcycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of consecutive edges (in the natural ordering of the edges) intersects in precisely ℓ vertices. We prove that if 1 ≤ ℓ < k and k − ℓ does not divide k then any kuniform hypergraph on n vertices with minimum degree at least nd k k− ` e(k−`) +o(n) contains a Hamilton ℓcycle. This confirms a conjecture of Hàn and Schacht. Together with results of Rödl, Ruciński and Szemerédi, our result asymptotically determines the minimum degree which forces an `cycle for any ` with 1 ≤ ℓ < k.
A geometric theory for hypergraph matching
 Memoirs of the American Mathematical Society
, 1908
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Graph removal lemmas
 SURVEYS IN COMBINATORICS
, 2013
"... The graph removal lemma states that any graph on n vertices with o(nv(H)) copies of a fixed graph H may be made Hfree by removing o(n²) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and com ..."
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Cited by 10 (4 self)
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The graph removal lemma states that any graph on n vertices with o(nv(H)) copies of a fixed graph H may be made Hfree by removing o(n²) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and computer science. In this survey we discuss these lemmas, focusing in particular on recent improvements to their quantitative aspects.
On the Ramsey number of sparse 3graphs
, 2008
"... Abstract. We consider a hypergraph generalization of a conjecture of Burr ..."
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Cited by 8 (0 self)
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Abstract. We consider a hypergraph generalization of a conjecture of Burr
A hypergraph regularity method for generalized Turán problems
 Random Structures & Algorithms
"... ABSTRACT: We describe a method that we believe may be foundational for a comprehensive theory of generalized Turán problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particu ..."
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ABSTRACT: We describe a method that we believe may be foundational for a comprehensive theory of generalized Turán problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particular configuration but also showing that these copies are evenly distributed. We demonstrate the power of the method by proving a conjecture of Mubayi on the codegree threshold of the Fano plane, that is, any 3graph on n vertices for which every pair of vertices is contained in more than n/2 edges must contain a Fano plane, for n sufficiently large. For projective planes over fields of odd size q we show that the codegree threshold is between n/2 − q + 1 and n/2, but for PG2(4) we find the somewhat surprising phenomenon that the threshold is less than (1/2 − )n for some small > 0. We conclude by setting out a program for future developments of this method to tackle other problems.