# Dynamic Ramsey Theory of Mechanical Systems Forming a Complete Graph and Vibrations of Cyclic Compounds

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## Abstract

**:**

## 1. Introduction

## 2. Ramsey Theory and Vibrations of Cyclic Molecules

#### 2.1. Ramsey Theory for the System Interconnected by Two Kinds of Ideal Springs

#### 2.2. Direct and Inverse Ramsey Networks of Ideal Springs

#### 2.3. Ramsey Theory for the System of Vibrating Masses Partially Connected by the Ideal Springs

#### 2.4. Multi-Color Systems Built of Ideal Springs

#### 2.5. Ramsey Model of Viscoelasticity

#### 2.6. Ramsey Theory for Vibrations of Systems in Which Entropy Elasticity Is Present

## 3. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**Modes of the vibrations occurring within a system built from three equal masses m (

**a**) and springs ${k}_{2}$. (

**b**) Springs form an equilateral triangle. Purple arrows show the displacement of the masses.

## References

- Ramsey, F.P. On a Problem of Formal Logic. In Classic Papers in Combinatorics. Modern Birkhäuser Classics; Gessel, I., Rota, G.C., Eds.; Birkhäuser: Boston, MA, USA, 2009; pp. 264–286. [Google Scholar] [CrossRef]
- Katz, M.; Reimann, J. An Introduction to Ramsey Theory: Fast Functions, Infinity, and Metamathematics; Student Mathematical Library, American Mathematical Society: Providence, RI, USA, 2018; Volume 87, pp. 1–34. [Google Scholar]
- Di Nasso, M.; Goldbring, I.; Lupini, M. Nonstandard Methods in Combinatorial Number Theory, Lecture Notes in Mathematics; Springer-Verlag: Berlin, Germany, 2019; Volume 2239. [Google Scholar]
- Graham, R.L.; Spencer, J.H. Ramsey Theory. Sci. Am.
**1990**, 7, 112–117. [Google Scholar] [CrossRef] - Graham, R.; Butler, S. Rudiments of Ramsey Theory, 2nd ed.; American Mathematical Society: Providence, RI, USA, 2015; pp. 7–46. [Google Scholar]
- Graham, R.L.; Rothschild, B.L.; Spencer, J.H. Ramsey Theory, 2nd ed.; Wiley-Interscience Series in Discrete Mathematics and Optimization; John Wiley & Sons, Inc.: New York, NY, USA, 1990; pp. 10–110. [Google Scholar]
- Roberts, F.S. Applications of Ramsey theory. Discret. Appl. Math.
**1984**, 9, 251–261. [Google Scholar] [CrossRef] - Wouters, J.; Giotis, A.; Kang, R.; Schuricht, D.; Fritz, L. Lower bounds for Ramsey numbers as a statistical physics problem. J. Stat. Mech.
**2022**, 2022, 332. [Google Scholar] [CrossRef] - Shvalb, N.; Frenkel, M.; Shoval, S.; Bormashenko, E. Ramsey theory and thermodynamics. Heliyon
**2023**, 9, e13561. [Google Scholar] [CrossRef] [PubMed] - Shvalb, N.; Frenkel, M.; Shoval, S.; Bormashenko, E. Universe as a Graph (Ramsey Approach to Analysis of Physical Systems). World J. Phys.
**2023**, 1, 1–24. [Google Scholar] - Randic, M. On Characterization of Cyclic Structures. J. Chem. Inf. Comput. Sci.
**1997**, 37, 1063–1071. [Google Scholar] [CrossRef] - Garcia-Domenech, R.; Galvez, J.; de Julián-Ortiz, J.V.; Pogliani, L. Some New Trends in Chemical Graph Theory. Chem. Rev.
**2008**, 108, 1127–1169. [Google Scholar] [CrossRef] [PubMed] - Zhou, X.; Li, K.; Goodman, M.; Sallam, A. A Novel Approach for the Classical Ramsey Number Problem on DNA-Based Supercomputing. Match Commun. Math. Comput. Chem.
**2011**, 66, 347–370. [Google Scholar] - Goldstein, H.; Poole, C.P.; Safko, J.L. Classical Mechanics, 3rd ed.; Addison-Wesley: Menlo Park, CA, USA, 2001; Chapter 6. [Google Scholar]
- Collins, J.J.; Stewart, I. A group-theoretic approach to rings of coupled biological oscillators. Biol. Cybern.
**1994**, 71, 95–103. [Google Scholar] [CrossRef] [PubMed] - Qin, W.-X.; Zhang, P.-L. Discrete rotating waves in a ring of coupled mechanical oscillators with strong damping. J. Math. Phys.
**2009**, 50, 52701. [Google Scholar] [CrossRef] - Lievens, S.; Stoilova, N.I.; Van der Jeugt, J. A linear chain of interacting harmonic oscillators: Solutions as a Wigner Quantum System. J. Phys. Conf. Ser.
**2008**, 128, 12028. [Google Scholar] [CrossRef] - Vinogradov, G.V.; Malkin, A.Y. Rheology of Polymers: Viscoelasticity and Flow of Polyme; USSR Mir: Moscow, Russia, 1980. [Google Scholar]
- Sánchez, M.A.G.; Giraldo-Vásquez, D.H.; Sánchez, R.M. Rheometric, transient, and cyclic tests to assess the viscoelastic behavior of natural rubber-based compounds used for rubber bearings. Mater. Today Commun.
**2020**, 22, 100815. [Google Scholar] [CrossRef] - Breńkacz, L.; Bagiński, P.; Korbicz, J.K. Vibration damping of the anti-vibration platform intended for use in combination with audio/music devices. J. Vibroeng.
**2020**, 22, 578–593. [Google Scholar] [CrossRef] - Christensen, R. Theory of Viscoelasticity: An Introduction; Elsevier: Amsterdam, The Netherlands, 2012; Chapter 1. [Google Scholar]
- Rubinstein, M.; Colby, R.H. Polymer Physcis; Oxford University Press: Oxford, UK, 2003; Chapter 2. [Google Scholar]
- Braun, M.; Lansky, Z.; Hilitski, F.; Dogic, Z.; Diez, S. Entropic forces drive contraction of cytoskeletal networks. BioEssays
**2016**, 38, 474–481. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Cyclic chemical compound represented by a mechanical system forming a complete graph. The system is built of identical masses m interconnected by two kinds of springs ${k}_{1}$ (red ones) and ${k}_{2}\left(\mathrm{green}\mathrm{ones}\right)$.

**Figure 2.**Equal masses m forming a regular hexagon are interconnected by two kinds of ideal springs denoted, ${k}_{1}$ (green edges) and ${k}_{2}$ (red edges). Two equilateral triangles “153” and “246” are recognized.

**Figure 3.**Five equal masses m forming a regular pentagon are interconnected by two kinds of ideal springs denoted ${k}_{1}$ (green edges) and ${k}_{2}$ (red edges). No triangles are formed in the spring network. The Ramsey number $R\left(3,3\right)=6>5$.

**Figure 4.**3D systems built of two tetrahedrons “1234” and “1235” are depicted. Masses placed in the vertices of the tetrahedrons are connected by two kinds of springs. The green (${k}_{1})$ and red (${k}_{2}$) links denote the springs. Triangle “123” is located in the plane $\left(XOY\right)$.

**Figure 5.**Symmetrical system of six-point masses m partially interconnected by identical ideal springs k. Masses labeled ”14”, “45” and “36” are not connected by springs. Masses connected by springs are connected by black links; masses which are not joint with the springs are connected by red links. Black triangles “156” and “234” are present in the graph.

**Figure 6.**The Ramsey model of viscoelastic body is presented. Point masses ${m}_{i}\left(i=1\dots 6\right)$ are connected by ideal springs k or with viscous elements, quantified by viscosity $\eta $. Viscous joints are shown in red whereas elastic joints are connected by black links.

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**MDPI and ACS Style**

Shvalb, N.; Frenkel, M.; Shoval, S.; Bormashenko, E.
Dynamic Ramsey Theory of Mechanical Systems Forming a Complete Graph and Vibrations of Cyclic Compounds. *Dynamics* **2023**, *3*, 272-281.
https://doi.org/10.3390/dynamics3020016

**AMA Style**

Shvalb N, Frenkel M, Shoval S, Bormashenko E.
Dynamic Ramsey Theory of Mechanical Systems Forming a Complete Graph and Vibrations of Cyclic Compounds. *Dynamics*. 2023; 3(2):272-281.
https://doi.org/10.3390/dynamics3020016

**Chicago/Turabian Style**

Shvalb, Nir, Mark Frenkel, Shraga Shoval, and Edward Bormashenko.
2023. "Dynamic Ramsey Theory of Mechanical Systems Forming a Complete Graph and Vibrations of Cyclic Compounds" *Dynamics* 3, no. 2: 272-281.
https://doi.org/10.3390/dynamics3020016