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Regular Partitions of Graphs
Abstract : A crucial lemma in recent work of the author (showing that k-term arithmetic progression-free sets of integers must have density zero) stated (approximately) that any large bipartite graphExpand
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On sets of integers containing k elements in arithmetic progression
In 1926 van der Waerden [13] proved the following startling theorem : If the set of integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmeticExpand
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Extremal problems in discrete geometry
TLDR
We establish several theorems involving configurations of points and lines in the Euclidean plane. Expand
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Blow-up Lemma
TLDR
Regular pairs behave like complete bipartite graphs from the point of view of bounded degree subgraphs. Expand
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A Dirac-Type Theorem for 3-Uniform Hypergraphs
TLDR
A Hamiltonian cycle in a 3-uniform hypergraph is a cyclic ordering of the vertices in which every three consecutive vertices form an edge. Expand
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A Note on Ramsey Numbers
TLDR
We prove R(3, x) cx 2 ln x and, for each k ⩾ 3, R(k,x) c k x k − 2 asymptotically in x . Expand
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Perfect matchings in large uniform hypergraphs with large minimum collective degree
We define a perfect matching in a k-uniform hypergraph H on n vertices as a set of @?n/k@? disjoint edges. Let @d"k"-"1(H) be the largest integer d such that every (k-1)-element set of vertices of HExpand
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O(n LOG n) SORTING NETWORK.
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On sums and products of integers
Let 1 ≦ a 1 <... <a n be a sequence of integers Consider the integers of the form $$a_i + a_j ,\,a_i a_j ,\,1 \leqq i \leqq j \leqq n.$$ (1)
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Storing a sparse table with O(1) worst case access time
TLDR
We describe a data structure for representing a set of n items from a universe of m items, which uses space n+o(n) and accommodates membership queries in constant time. Expand
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