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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 graph
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 arithmetic
Extremal problems in discrete geometry
TLDR
Several theorems involving configurations of points and lines in the Euclidean plane are established, including one that shows that there is an absolute constantc3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc3n of the lines determined by then points.
Blow-up Lemma
Regular pairs behave like complete bipartite graphs from the point of view of bounded degree subgraphs.
A Note on Ramsey Numbers
A Dirac-Type Theorem for 3-Uniform Hypergraphs
TLDR
An approximate and asymptotic version of an analogue of Dirac's celebrated theorem for graphs is proved: for each γ>0 there exists n0 such that every 3-uniform hypergraph on n_0 vertices, in which each pair of vertices belongs to at least $(1/2+\gamma)n$ edges, contains a Hamiltonian cycle.
O(n LOG n) SORTING NETWORK.
Storing a sparse table with O(1) worst case access time
TLDR
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 and is easy to implement.
On the probability that a random ±1-matrix is singular
We report some progress on the old problem of estimating the probability, Pn, that a random n× n ± 1 matrix is singular: Theorem. There is a positive constant ε for which Pn < (1− ε)n. This is a
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