1,338 research outputs found
On Ramsey numbers of complete graphs with dropped stars
Let be the smallest integer such that for any -coloring (say,
red and blue) of the edges of , , there is either a red
copy of or a blue copy of . Let be the complete graph on
vertices from which the edges of are dropped. In this note we
present exact values for and new upper bounds
for in numerous cases. We also present some results for
the Ramsey number of Wheels versus .Comment: 9 pages ; 1 table in Discrete Applied Mathematics, Elsevier, 201
Two conjectures in Ramsey-Tur\'an theory
Given graphs , a graph is -free if
there is a -edge-colouring with no monochromatic
copy of with edges of colour for each . Fix a function
, the Ramsey-Tur\'an function is the
maximum number of edges in an -vertex -free graph with
independence number at most . We determine for and sufficiently small , confirming a
conjecture of Erd\H{o}s and S\'os from 1979. It is known that
has a phase transition at . However, the values of was not
known. We determined this value by proving , answering a question of Balogh, Hu and Simonovits.
The proofs utilise, among others, dependent random choice and results from
graph packings.Comment: 20 pages, 2 figures, 2 pages appendi
On small Mixed Pattern Ramsey numbers
We call the minimum order of any complete graph so that for any coloring of
the edges by colors it is impossible to avoid a monochromatic or rainbow
triangle, a Mixed Ramsey number. For any graph with edges colored from the
above set of colors, if we consider the condition of excluding in the
above definition, we produce a \emph{Mixed Pattern Ramsey number}, denoted
. We determine this function in terms of for all colored -cycles
and all colored -cliques. We also find bounds for when is a
monochromatic odd cycles, or a star for sufficiently large . We state
several open questions.Comment: 16 page
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
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