129 research outputs found
Using Ramsey theory to measure unavoidable spurious correlations in Big Data
Given a dataset we quantify how many patterns must always exist in the
dataset. Formally this is done through the lens of Ramsey theory of graphs, and
a quantitative bound known as Goodman's theorem. Combining statistical tools
with Ramsey theory of graphs gives a nuanced understanding of how far away a
dataset is from random, and what qualifies as a meaningful pattern.
This method is applied to a dataset of repeated voters in the 1984 US
congress, to quantify how homogeneous a subset of congressional voters is. We
also measure how transitive a subset of voters is. Statistical Ramsey theory is
also used with global economic trading data to provide evidence that global
markets are quite transitive.Comment: 21 page
Combinatorial theorems relative to a random set
We describe recent advances in the study of random analogues of combinatorial
theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201
Popular progression differences in vector spaces II
Green used an arithmetic analogue of Szemer\'edi's celebrated regularity
lemma to prove the following strengthening of Roth's theorem in vector spaces.
For every , , and prime number , there is a least
positive integer such that if ,
then for every subset of of density at least there is
a nonzero for which the density of three-term arithmetic progressions with
common difference is at least . We determine for the
tower height of up to an absolute constant factor and an
additive term depending only on . In particular, if we want half the random
bound (so ), then the dimension required is a tower of
twos of height . It turns
out that the tower height in general takes on a different form in several
different regions of and , and different arguments are used
both in the upper and lower bounds to handle these cases.Comment: 34 pages including appendi
Positional Games
Positional games are a branch of combinatorics, researching a variety of
two-player games, ranging from popular recreational games such as Tic-Tac-Toe
and Hex, to purely abstract games played on graphs and hypergraphs. It is
closely connected to many other combinatorial disciplines such as Ramsey
theory, extremal graph and set theory, probabilistic combinatorics, and to
computer science. We survey the basic notions of the field, its approaches and
tools, as well as numerous recent advances, standing open problems and
promising research directions.Comment: Submitted to Proceedings of the ICM 201
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