173 research outputs found
On the asymptotic minimum number of monochromatic 3-term arithmetic progressions
Let V(n) be the minimum number of monochromatic 3-term arithmetic
progressions in any 2-coloring of {1,2,...,n}. We show that (1675/32768) n^2
(1+o(1)) <= V(n) <= (117/2192) n^2(1+o(1)). As a consequence, we find that V(n)
is strictly greater than the corresponding number for Schur triples (which is
(1/22) n^2 (1+o(1)). Additionally, we disprove the conjecture that V(n) =
(1/16) n^2(1+o(1)), as well as a more general conjecture.Comment: 9 pages. Revised version fixes formatting errors (same text
Partition regularity and multiplicatively syndetic sets
We show how multiplicatively syndetic sets can be used in the study of
partition regularity of dilation invariant systems of polynomial equations. In
particular, we prove that a dilation invariant system of polynomial equations
is partition regular if and only if it has a solution inside every
multiplicatively syndetic set. We also adapt the methods of Green-Tao and
Chow-Lindqvist-Prendiville to develop a syndetic version of Roth's density
increment strategy. This argument is then used to obtain bounds on the Rado
numbers of configurations of the form .Comment: 29 pages. v3. Referee comments incorporated, accepted for publication
in Acta Arithmetic
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
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