Open Problems Column Edited by William Gasarch This Issue's Column!

2021 ◽  
Vol 51 (4) ◽  
pp. 30-46
Author(s):  
William Gasarch
Keyword(s):  
Elementary Proof ◽  
Ramsey Theory ◽  
Open Problems ◽  
General Theme ◽  
Open Questions ◽  
Ramsey Type

In this column we state a class of open problems in Ramsey Theory. The general theme is to take Ramsey-type statements that are false and weaken them by allowing the homogenous set to use more than one color. This concept is not new, and the theorems we state and/or prove are not new; however, the open questions that request easier proofs of the known theorems (or weaker versions) may be new. We use the phrase an elementary proof. This is not meant to be a technical or rigorous term. What we really mean is a proof that can be taught in an undergraduate combinatorics course. A good example of what we mean is the proof of Theorem 9.3.

Open Problems in Euclidean Ramsey Theory

Ramsey Theory ◽  
2011 ◽  
pp. 115-120
Author(s):  
Ron Graham ◽  
Eric Tressler
Keyword(s):  
Ramsey Theory ◽  
Open Problems ◽  
Euclidean Ramsey Theory

Artificial Life as a Tool for Biological Inquiry

1993 ◽  
Vol 1 (1_2) ◽  
pp. 1-13 ◽  
Author(s):  
Charles Taylor ◽  
David Jefferson
Keyword(s):  
Cultural Evolution ◽  
Artificial Life ◽  
Living Systems ◽  
Grand Challenge ◽  
Open Problems ◽  
Maintenance Of Sex ◽  
Open Questions ◽  
Shifting Balance ◽  
The Origin Of Life ◽  
Natural Life

Artificial life embraces those human-made systems that possess some of the key properties of natural life. We are specifically interested in artificial systems that serve as models of living systems for the investigation of open questions in biology. First we review some of the artificial life models that have been constructed with biological problems in mind, and classify them by medium (hardware, software, or “wetware”) and by level of organization (molecular, cellular, organismal, or population). We then describe several “grand challenge” open problems in biology that seem especially good candidates to benefit from artificial life studies, including the origin of life and self-organi- zation, cultural evolution, origin and maintenance of sex, shifting balance in evolution, the relation between fitness and adaptedness, the structure of ecosystems, and the nature of mind.

scholarly journals On-line Ramsey Theory

10.37236/1810 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
J. A. Grytczuk ◽  
M. Hałuszczak ◽  
H. A. Kierstead
Keyword(s):  
Planar Graphs ◽  
Ramsey Theory ◽  
Winning Strategy ◽  
Fixed Target ◽  
Outerplanar Graphs ◽  
On Line ◽  
Ramsey Type ◽  
Colorable Graph ◽  
Monochromatic Copy

The Ramsey game we consider in this paper is played on an unbounded set of vertices by two players, called Builder and Painter. In one move Builder introduces a new edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed target graph $H$, keeping the constructed graph in a prescribed class ${\cal G}$. The main problem is to recognize the winner for a given pair $H,{\cal G}$. In particular, we prove that Builder has a winning strategy for any $k$-colorable graph $H$ in the game played on $k$-colorable graphs. Another class of graphs with this strange self-unavoidability property is the class of forests. We show that the class of outerplanar graphs does not have this property. The question of whether planar graphs are self-unavoidable is left open. We also consider a multicolor version of Ramsey on-line game. To extend our main result for $3$-colorable graphs we introduce another Ramsey type game, which seems interesting in its own right.

Open problems

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Symplectic Topology ◽  
Open Problems ◽  
Open Questions

The final chapter of this book discusses some open questions and conjectures that either have served as guiding lights or have emerged in the study of symplectic topology over the last quarter of a century. The wide variety of problems, though inevitably incomplete, provides a snapshot of where the field is at the time of writing.

Convex Polynomial Approximation in the Uniform Norm: Conclusion

2005 ◽  
Vol 57 (6) ◽  
pp. 1224-1248 ◽  
Author(s):  
K. A. Kopotun ◽  
D. Leviatan ◽  
I. A. Shevchuk
Keyword(s):  
Convex Function ◽  
Polynomial Approximation ◽  
Uniform Norm ◽  
Degree Of Approximation ◽  
Modulus Of Smoothness ◽  
Jackson Type ◽  
Algebraic Polynomials ◽  
Type Estimate ◽  
Open Problems ◽  
Open Questions

AbstractEstimating the degree of approximation in the uniform norm, of a convex function on a finite interval, by convex algebraic polynomials, has received wide attention over the last twenty years. However, while much progress has been made especially in recent years by, among others, the authors of this article, separately and jointly, there have been left some interesting open questions. In this paper we give final answers to all those open problems. We are able to say, for each r-th differentiable convex function, whether or not its degree of convex polynomial approximation in the uniform norm may be estimated by a Jackson-type estimate involving the weighted Ditzian–Totik kth modulus of smoothness, and how the constants in this estimate behave. It turns out that for some pairs (k, r) we have such estimate with constants depending only on these parameters. For other pairs the estimate is valid, but only with constants that depend on the function being approximated, while there are pairs for which the Jackson-type estimate is, in general, invalid.

scholarly journals A Euclidean Ramsey Result in the Plane

10.37236/7148 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Sergei Tsaturian
Keyword(s):  
Elementary Proof ◽  
Ramsey Theory ◽  
Unit Distance ◽  
Euclidean Ramsey Theory

An old question in Euclidean Ramsey theory asks, if the points in the plane are red-blue coloured, does there always exist a red pair of points at unit distance or five blue points in line separated by unit distances? An elementary proof answers this question in the affirmative.

Rainbow Arithmetic Progressions and Anti-Ramsey Results

2003 ◽  
Vol 12 (5-6) ◽  
pp. 599-620 ◽  
Author(s):  
V Jungic ◽  
J Licht ◽  
M Mahdian ◽  
J Nesetril ◽  
R Radoicic
Keyword(s):  
Arithmetic Progression ◽  
Ramsey Theory ◽  
Arithmetic Progressions ◽  
Natural Numbers ◽  
Colour Class ◽  
Ramsey Type ◽  
General Perspective

The van der Waerden theorem in Ramsey theory states that, for every k and t and sufficiently large N, every k-colouring of [N] contains a monochromatic arithmetic progression of length t. Motivated by this result, Radoičić conjectured that every equinumerous 3-colouring of [3n] contains a 3-term rainbow arithmetic progression, i.e., an arithmetic progression whose terms are coloured with distinct colours. In this paper, we prove that every 3-colouring of the set of natural numbers for which each colour class has density more than 1/6, contains a 3-term rainbow arithmetic progression. We also prove similar results for colourings of . Finally, we give a general perspective on other anti-Ramsey-type problems that can be considered.

Discrimination

2017 ◽  
Author(s):  
Frej Klem Thomsen
Keyword(s):  
Moral Status ◽  
Statistical Discrimination ◽  
Differential Treatment ◽  
Clear Distinction ◽  
Inequality Of Opportunity ◽  
Open Problems ◽  
Open Questions ◽  
Direct Discrimination ◽  
Indirect Discrimination ◽  
Definition Of

The conceptualization and moral analysis of discrimination constitutes a burgeoning theoretical field, with a number of open problems and a rapidly developing literature. A central problem is how to define discrimination, both in its most basic direct sense and in the most prominent variations. A plausible definition of the basic sense of the word understands discrimination as disadvantageous differential treatment of two groups that is in some respect caused by the properties that distinguish the groups, but open questions remain on whether discrimination should be restricted to concern only particular groups, as well as on whether it is best conceived as a descriptive or a moralized concept. Furthermore, since this understanding limits direct discrimination to cases of differential treatment, it requires that we be able to draw a clear distinction between equal and differential treatment, a task that is less simple than it may appear, but that is helpful in clarifying indirect discrimination and statistical discrimination. The second major problem in theorizing discrimination is explaining what makes discrimination morally wrong. On this issue, there are four dominant contemporary answers: the valuational and expressive disrespect accounts, which hold that discrimination is wrong when and if the discriminator misestimates or expresses a misestimate of the moral status of the discriminatee; the unfairness account, which holds that discrimination is wrong when and if the discriminator unfairly increases inequality of opportunity; and the harm account, which holds that discrimination is wrong when and if the discriminator harms the discriminatee. Each of these accounts, however, faces important challenges in simultaneously providing a persuasive theoretical account and matching our intuitions about cases of impermissible discrimination.

scholarly journals OPEN QUESTIONS ABOUT RAMSEY-TYPE STATEMENTS IN REVERSE MATHEMATICS

2016 ◽  
Vol 22 (2) ◽  
pp. 151-169 ◽  
Author(s):  
LUDOVIC PATEY
Keyword(s):  
Reverse Mathematics ◽  
Ramsey's Theorem ◽  
The Past ◽  
Open Questions ◽  
Ramsey’S Theorem ◽  
Ramsey Type ◽  
Infinite Set

AbstractRamsey’s theorem states that for any coloring of then-element subsets of ℕ with finitely many colors, there is an infinite setHsuch that alln-element subsets ofHhave the same color. The strength of consequences of Ramsey’s theorem has been extensively studied in reverse mathematics and under various reducibilities, namely, computable reducibility and uniform reducibility. Our understanding of the combinatorics of Ramsey’s theorem and its consequences has been greatly improved over the past decades. In this paper, we state some questions which naturally arose during this study. The inability to answer those questions reveals some gaps in our understanding of the combinatorics of Ramsey’s theorem.

scholarly journals Electromagnetic counterparts of compact binary mergers

2021 ◽  
Vol 87 (1) ◽  
Author(s):  
Stefano Ascenzi ◽  
Gor Oganesyan ◽  
Marica Branchesi ◽  
Riccardo Ciolfi
Keyword(s):  
Gravitational Waves ◽  
High Energy ◽  
Design Sensitivity ◽  
Compact Objects ◽  
Open Problems ◽  
Binary Neutron Star ◽  
Compact Binary ◽  
Open Questions ◽  
Binary Mergers ◽  
Electromagnetic Signals

The first detection of a binary neutron star merger through gravitational waves and photons marked the dawn of multimessenger astronomy with gravitational waves, and it greatly increased our insight in different fields of astrophysics and fundamental physics. However, many open questions on the physical process involved in a compact binary merger still remain and many of these processes concern plasma physics. With the second generation of gravitational wave interferometers approaching their design sensitivity, the new generation under design study and new X-ray detectors under development, the high energy universe will become more and more a unique laboratory for our understanding of plasma in extreme conditions. In this review, we discuss the main electromagnetic signals expected to follow the merger of two compact objects highlighting the main physical processes involved and some of the most important open problems in the field.

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