1,620 research outputs found
Partition regularity of a system of De and Hindman
We prove that a certain matrix, which is not image partition regular over R
near zero, is image partition regular over N. This answers a question of De and
Hindman.Comment: 7 page
Random walks on quasirandom graphs
Let G be a quasirandom graph on n vertices, and let W be a random walk on G
of length alpha n^2. Must the set of edges traversed by W form a quasirandom
graph? This question was asked by B\"ottcher, Hladk\'y, Piguet and Taraz. Our
aim in this paper is to give a positive answer to this question. We also prove
a similar result for random embeddings of trees.Comment: 19 pages, 2 figure
Partition regularity with congruence conditions
An infinite integer matrix A is called image partition regular if, whenever
the natural numbers are finitely coloured, there is an integer vector x such
that Ax is monochromatic. Given an image partition regular matrix A, can we
also insist that each variable x_i is a multiple of some given d_i? This is a
question of Hindman, Leader and Strauss.
Our aim in this short note is to show that the answer is negative. As an
application, we disprove a conjectured equivalence between the two main forms
of partition regularity, namely image partition regularity and kernel partition
regularity.Comment: 5 page
Maximum hitting for n sufficiently large
For a left-compressed intersecting family \A contained in [n]^(r) and a set X
contained in [n], let \A(X) = {A in \A : A intersect X is non-empty}. Borg
asked: for which X is |\A(X)| maximised by taking \A to be all r-sets
containing the element 1? We determine exactly which X have this property, for
n sufficiently large depending on r.Comment: Version 2 corrects the calculation of the sizes of the set families
appearing in the proof of the main theorem. It also incorporates a number of
other smaller corrections and improvements suggested by the anonymous
referees. 7 page
Policy Forum: Examining the Relationship Between Vacant Health Director Positions & Public Health
Policy Forum: The Tobacco Surcharge & Sugar-Sweetened Beverage Taxes: Reconciling Equity and Targeted Public Health Interventions
Virginia is poised to repeal the tobacco surcharge, an ineffective policy that disproportionately harms low- and middle-income Virginians. (Small, 2023) The Patient Protection and Affordable Care Act (ACA), which otherwise strengthened health care access and equity, allows health insurers in the individual and small-group markets to charge smokers up to 50 percent higher premiums relative to nonsmokers. (ACA, 2010) The law’s financial assistance does not apply to this surcharge, forcing enrollees to bear the entire cost of the penalty. This provision was a compromise between the ACA’s drafters, most of whom opposed the surcharge, and the health insurance industry, which argued that insurers would need to raise premiums on all enrollees if they could not charge smokers higher premiums
Partition regularity without the columns property
A finite or infinite matrix A with rational entries is called partition
regular if whenever the natural numbers are finitely coloured there is a
monochromatic vector x with Ax=0. Many of the classical theorems of Ramsey
Theory may naturally be interpreted as assertions that particular matrices are
partition regular. In the finite case, Rado proved that a matrix is partition
regular if and only it satisfies a computable condition known as the columns
property. The first requirement of the columns property is that some set of
columns sums to zero.
In the infinite case, much less is known. There are many examples of matrices
with the columns property that are not partition regular, but until now all
known examples of partition regular matrices did have the columns property. Our
main aim in this paper is to show that, perhaps surprisingly, there are
infinite partition regular matrices without the columns property --- in fact,
having no set of columns summing to zero.
We also make a conjecture that if a partition regular matrix (say with
integer coefficients) has bounded row sums then it must have the columns
property, and prove a first step towards this.Comment: 13 page
Small sums of five roots of unity
Motivated by questions in number theory, Myerson asked how small the sum of 5
complex nth roots of unity can be. We obtain a uniform bound of O(n^{-4/3}) by
perturbing the vertices of a regular pentagon, improving to O(n^{-7/3})
infinitely often.
The corresponding configurations were suggested by examining exact minimum
values computed for n <= 221000. These minima can be explained at least in part
by selection of the best example from multiple families of competing
configurations related to close rational approximations
Distinguishing subgroups of the rationals by their Ramsey properties
A system of linear equations with integer coefficients is partition regular
over a subset S of the reals if, whenever S\{0} is finitely coloured, there is
a solution to the system contained in one colour class. It has been known for
some time that there is an infinite system of linear equations that is
partition regular over R but not over Q, and it was recently shown (answering a
long-standing open question) that one can also distinguish Q from Z in this
way.
Our aim is to show that the transition from Z to Q is not sharp: there is an
infinite chain of subgroups of Q, each of which has a system that is partition
regular over it but not over its predecessors. We actually prove something
stronger: our main result is that if R and S are subrings of Q with R not
contained in S, then there is a system that is partition regular over R but not
over S. This implies, for example, that the chain above may be taken to be
uncountable.Comment: 14 page
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