scholarly journals A quantitative Lovász criterion for Property B

2020 ◽  
Vol 29 (6) ◽  
pp. 956-960
Author(s):  
Asaf Ferber ◽  
Asaf Shapira
Keyword(s):  
Short Paper ◽  
Uniform Hypergraph ◽  
Quantitative Version ◽  
Single Vertex ◽  
Property B

AbstractA well-known observation of Lovász is that if a hypergraph is not 2-colourable, then at least one pair of its edges intersect at a single vertex. In this short paper we consider the quantitative version of Lovász’s criterion. That is, we ask how many pairs of edges intersecting at a single vertex should belong to a non-2-colourable n-uniform hypergraph. Our main result is an exact answer to this question, which further characterizes all the extremal hypergraphs. The proof combines Bollobás’s two families theorem with Pluhar’s randomized colouring algorithm.

scholarly journals Optimal Divisibility Conditions for Loose Hamilton Cycles in Random Hypergraphs

10.37236/2523 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Andrzej Dudek ◽  
Alan Frieze ◽  
Po-Shen Loh ◽  
Shelley Speiss
Keyword(s):  
Absolute Constant ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Single Vertex ◽  
Random Hypergraphs

In the random $k$-uniform hypergraph $H^{(k)}_{n,p}$ of order $n$, each possible $k$-tuple appears independently with probability $p$. A loose Hamilton cycle is a cycle of order $n$ in which every pair of consecutive edges intersects in a single vertex. It was shown by Frieze that if $p\geq c(\log n)/n^2$ for some absolute constant $c>0$, then a.a.s. $H^{(3)}_{n,p}$ contains a loose Hamilton cycle, provided that $n$ is divisible by $4$. Subsequently,  Dudek and Frieze extended this result for any uniformity $k\ge 4$, proving that if $p\gg (\log n)/n^{k-1}$, then $H^{(k)}_{n,p}$ contains a loose Hamilton cycle, provided that $n$ is divisible by $2(k-1)$. In this paper, we improve the divisibility requirement and show that in the above results it is enough to assume that $n$ is a multiple of $k-1$, which is best possible.

Coloring Four-uniform Hypergraphs on Nine Vertices

2015 ◽  
Vol 58 (2) ◽  
pp. 317-319
Author(s):  
Marek Kazimierz Lewkowicz
Keyword(s):  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Property B

AbstractEvery 4-uniform hypergraph on 9 vertices with at most 25 edges has property B. This gives the answer m9(4) = 26 to a question raised by Erdős in 1968.

scholarly journals Loose Hamilton Cycles in Random Uniform Hypergraphs

10.37236/535 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Andrzej Dudek ◽  
Alan Frieze
Keyword(s):  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Single Vertex ◽  
Uniform Hypergraphs

In the random $k$-uniform hypergraph $H_{n,p;k}$ of order $n$ each possible $k$-tuple appears independently with probability $p$. A loose Hamilton cycle is a cycle of order $n$ in which every pair of adjacent edges intersects in a single vertex. We prove that if $p n^{k-1}/\log n$ tends to infinity with $n$ then $$\lim_{\substack{n\to \infty\\ 2(k-1) |n}}\Pr(H_{n,p;k}\text{ contains a loose Hamilton cycle})=1.$$ This is asymptotically best possible.

scholarly journals The Ramsey Number of Fano Plane Versus Tight Path

10.37236/8374 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
József Balogh ◽  
Felix Christian Clemen ◽  
Jozef Skokan ◽  
Adam Zsolt Wagner
Keyword(s):  
Edge Coloring ◽  
Ramsey Number ◽  
Short Paper ◽  
Simple Construction ◽  
Uniform Hypergraph ◽  
Fano Plane ◽  
Blue Edge ◽  
Uniform Hypergraphs ◽  
Plane Versus

The hypergraph Ramsey number of two $3$-uniform hypergraphs $G$ and $H$, denoted by $R(G,H)$, is the least integer~$N$ such that every red-blue edge-coloring of the complete $3$-uniform hypergraph on $N$ vertices contains a red copy of $G$ or a blue copy of $H$. The Fano plane $\mathbb{F}$ is the unique 3-uniform hypergraph with seven edges on seven vertices in which every pair of vertices is contained in a unique edge. There is a simple construction showing that $R(H,\mathbb{F}) \ge 2(v(H)-1) + 1.$  Hypergraphs $H$ for which the equality holds are called $\mathbb{F}$-good. Conlon asked to determine all $H$ that are $\mathbb{F}$-good.In this short paper we make progress on this problem and prove that the tight path of length $n$ is $\mathbb{F}$-good.

The promises and perfidy of cryomicroscopy and analysis

1994 ◽  
Vol 52 ◽  
pp. 382-383
Author(s):  
Patrick Echlin
Keyword(s):  
Low Temperature ◽  
High Energy ◽  
Short Paper ◽  
List Type ◽  
Time Resolved ◽  
Energy Beam ◽  
Cellular Analysis ◽  
Dissolved Ions ◽  
Embedding Medium ◽  
Low Temperature Microscopy

The unusual title of this short paper and its accompanying tutorial is deliberate, because the intent is to investigate the effectiveness of low temperature microscopy and analysis as one of the more significant elements of the less interventionist procedures we can use to prepare, examine and analyse hydrated and organic materials in high energy beam instruments. The promises offered by all these procedures are well rehearsed and the litany of petitions and responses may be enunciated in the following mantra.Vitrified water can form the perfect embedding medium for bio-organic samples.Frozen samples provide an important, but not exclusive, milieu for the in situ sub-cellular analysis of the dissolved ions and electrolytes whose activities are central to living processes.The rapid conversion of liquids to solids provides a means of arresting dynamic processes and permits resolution of the time resolved interactions between water and suspended and dissolved materials.The low temperature environment necessary for cryomicroscopy and analysis, diminish, but alas do not prevent, the deleterious side effects of ionizing radiation.Sample contamination is virtually eliminated.

Multi-parameter Singular Integrals. (AM-189)

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Partial Differential Equations ◽  
Graduate Students ◽  
General Theory ◽  
Differential Equations ◽  
Singular Integrals ◽  
Main Tool ◽  
Classical Theorem ◽  
Quantitative Version ◽  
Linear Partial Differential Equations

This book develops a new theory of multi-parameter singular integrals associated with Carnot–Carathéodory balls. The book first details the classical theory of Calderón–Zygmund singular integrals and applications to linear partial differential equations. It then outlines the theory of multi-parameter Carnot–Carathéodory geometry, where the main tool is a quantitative version of the classical theorem of Frobenius. The book then gives several examples of multi-parameter singular integrals arising naturally in various problems. The final chapter of the book develops a general theory of singular integrals that generalizes and unifies these examples. This is one of the first general theories of multi-parameter singular integrals that goes beyond the product theory of singular integrals and their analogs. This book will interest graduate students and researchers working in singular integrals and related fields.

Frontiers in Complex Dynamics

2014 ◽  
Author(s):  
Araceli Bonifant ◽  
Misha Lyubich ◽  
Scott Sutherland
Keyword(s):  
Historical Perspective ◽  
Complex Dynamics ◽  
Short Paper ◽  
Combinatorial Group Theory ◽  
Entropy Theory ◽  
Holomorphic Dynamics ◽  
Eightieth Birthday ◽  
Several Variables ◽  
Fields Medal ◽  
Combinatorial Group

John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing.

Social Attitudes and Empathy in the Weather Enterprise

2018 ◽  
Author(s):  
Matthew J. Bolton
Keyword(s):  
Social Attitudes ◽  
Short Paper

A short paper on social attitudes in the American weather enterprise, accepted for publication in the July 2018 newsletter of the National Weather Association.

An Expert Ranked List of Pro-nature Conservation Behaviours for Public Use

2019 ◽  
Author(s):  
Lea Barbett ◽  
Edward Stupple ◽  
Michael Sweet ◽  
Miles Richardson
Keyword(s):  
Nature Conservation ◽  
General Population ◽  
Mass Extinction ◽  
Negative Impact ◽  
Ecological Impact ◽  
Human Action ◽  
Short Paper ◽  
The Public ◽  
The Uk ◽  
Ranked List

The planet is facing an anthropogenic mass extinction of wildlife, which will have a grave impact on the environment and humans. Widespread human action is needed to minimize the negative impact of humans on biodiversity and support the restoration of wildlife. In order to find effective ways to promote pro-nature conservation behaviours to the general population, there is a need to provide a list of behaviours which will have worthwhile ecological impact and are worth encouraging. In a novel collaboration between psychologists and ecologists, 70 experts from practical and academic conservation backgrounds were asked to review and rate 48 conservation related behaviours. According to their judgement, this short paper presents a ranked list of pro-nature conservation behaviours for the public in the UK and similar landscapes. This includes behaviours people can engage in in their homes, their gardens, on their land, and in their roles as citizens.

Explaining spreadsheets with spreadsheets (short paper)

2020 ◽  
Vol 53 (9) ◽  
pp. 161-167
Author(s):  
Jácome Cunha ◽  
Mihai Dan ◽  
Martin Erwig ◽  
Danila Fedorin ◽  
Alex Grejuc
Keyword(s):  
Short Paper
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