# The Cyclic Triangle-Free Process

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminary

#### 2.1. The Ramsey Number $R(s,t)$

#### 2.2. $R(3,t)$ and the Triangle-Free Process

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

## 3. The Cyclic Triangle-Free Process

**Theorem**

**4.**

## 4. The Sizes of Parameter Sets of Cyclic Graphs Obtained by the Cyclic Triangle-Free Process

#### 4.1. Computation on the Sizes of Parameter Sets of Cyclic Graphs in Large Cases

#### 4.2. A Simple Lower Bound on the Sizes of Parameter Sets

**Theorem**

**5.**

#### 4.3. More Computation on the Sizes of Parameter Sets of Cyclic Graphs

## 5. Independence Numbers and Lower Bounds for Small $R(3,t)$

**Theorem**

**6.**

## 6. Conclusions and Problems

**Problem**

**1.**

**Problem**

**2.**

**Problem**

**3.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Ramsey, F.P. On a problem of formal logic. Proc. Lond. Math. Soc.
**1930**, s2–s30, 264–286. [Google Scholar] [CrossRef] - Codenotti, B.; Gerace, I.; Vigna, S. Hardness results and spectral techniques for combinatorial problems on circulant graphs. Linear Algebra Appl.
**1998**, 285, 123–142. [Google Scholar] [CrossRef] [Green Version] - Bohman, T.; Keevash, P. Dynamic concentration of the triangle-free process. arXiv
**2013**, arXiv:1302.5963. [Google Scholar] - Pontiveros, G.F.; Griffiths, S.; Morris, R. The triangle-free process and R(3,k). arXiv
**2013**, arXiv:1302.6279. [Google Scholar] - Bohman, T.; Keevash, P. The early evolution of the H-free process. Invent. Math.
**2010**, 181, 291–336. [Google Scholar] [CrossRef] - Krivelevich, M.; Kwan, M.; Loh, P.S.; Sudakov, B. The random k-matching-free process. arXiv
**2018**, arXiv:1708.01054v2. [Google Scholar] [CrossRef] - Tran, T. On the structure of large sum-free sets of integers. Isr. J. Math.
**2018**, 228, 249–292. [Google Scholar] [CrossRef] [Green Version] - Haviv, I.; Levy, D. Symmetric complete sum-free sets in cyclic groups. Isr. J. Math.
**2018**, 227, 931–956. [Google Scholar] [CrossRef] [Green Version] - Radziszowski, S.P. Small Ramsey numbers. Electr. J. Comb.
**2017**, DS1, 1–104. [Google Scholar] - Erdős, P. Graph theory and probability II. Can. J. Math.
**1961**, 13, 346–352. [Google Scholar] [CrossRef] - Ajtai, M.; Komlós, J.; Szemerédi, E. A note on Ramsey numbers. J. Comb. Theory Ser. A
**1980**, 29, 354–360. [Google Scholar] [CrossRef] [Green Version] - Soifer, A. (Ed.) Ramsey Theory: Yesterday, Today, and Tomorrow, 1st ed.; Birkhäuser: Boston, MA, USA, 2011. [Google Scholar]
- Shearer, J.B. A note on the independence number of triangle-free graphs. Discret. Math.
**1983**, 46, 83–87. [Google Scholar] [CrossRef] [Green Version] - Kim, J.H. The Ramsey number R(3,t) has order of magnitude t
^{2}/logt. Random Struct. Algorithms**1995**, 7, 173–207. [Google Scholar] [CrossRef] - Wu, K.; Su, W.; Luo, H.; Xu, X. New lower bounds for seven classical Ramsey numbers R(3,q). Appl. Math. Lett.
**2009**, 22, 365–368. [Google Scholar] [CrossRef] - Wu, K.; Su, W.; Luo, H.; Xu, X. A generalization of generalized Paley graphs and new lower bounds for R(3,q). Electr. J. Comb.
**2010**, 17, N25:1–N25:10. [Google Scholar] - Li, M.; Li, Y. Ramsey numbers and triangle-free Cayley graphs. J. Tongji Univ. (Nat. Sci.)
**2015**, 43, 1750–1752. (In Chinese) [Google Scholar] - Deng, F.; Shao, Z.; Xu, X. An algorithm for finding optimal lower bounds on Ramsey numbers based on cyclic graphs. J. Comput. Theor. Nanosci.
**2012**, 9, 1603–1605. [Google Scholar] [CrossRef] - Nenadov, R.; Person, Y.; Skoric, N.; Steger, A. An algorithmic framework for obtaining lower bounds for random Ramsey problems. J. Comb. Theory Ser. B
**2017**, 124, 1–38. [Google Scholar] [CrossRef] [Green Version] - Burr, S.A.; Erdős, P.; Faudree, R.J.; Schelp, R.H. On the difference between consecutive Ramsey numbers. Util. Math.
**1989**, 35, 115–118. [Google Scholar] - Xu, X.; Xie, Z.; Radziszowski, S.P. A constructive approach for the lower bounds on the Ramsey numbers R(s,t). J. Graph Theory
**2004**, 47, 231–239. [Google Scholar] - Chung, F.R.K.; Cleve, R.; Dagum, P. A note on constructive lower bounds for the Ramsey numbers R(3,t). J. Comb. Theory Ser. B
**1993**, 57, 150–155. [Google Scholar] [CrossRef] - Xu, X.; Liang, M.; Luo, H. Some Unsolved Problems and Results in Ramsey Theory; Walter de Gruyter GmbH: Berlin, Germany; Boston, MA, USA; University of Science and Technology of China Press: Hefei, China, 2018. [Google Scholar]
- Fredricksen, H.; Sweet, M.M. Symmetric sum-free partitions and lower bounds for Schur numbers. Electr. J. Comb.
**2000**, 7, R32:1–R32:9. [Google Scholar] - Xu, X.; Xie, Z.; Exoo, G.; Radziszowski, S.S.P. Constructive lower bounds on classical multicolor Ramsey numbers. Electr. J. Comb.
**2004**, 11, R35:1–R35:24. [Google Scholar] - Xu, X.; Radziszowski, S.P. Bounds on Shannon capacity and Ramsey numbers from product of graphs. IEEE Trans. Inf. Theory
**2013**, 59, 4767–4770. [Google Scholar] - Zhu, R.; Xu, X.; Radziszowski, S.P. A small step forwards on the Erdős-Sós problem concerning the Ramsey numbers R(3,k). Discret. Appl. Math.
**2016**, 214, 216–221. [Google Scholar] [CrossRef]

Order | Average | Range | Order | Average | Range |
---|---|---|---|---|---|

122 | 14.33 | 11–31 | 290 | 25.44 | 20–49 |

197 | 18.95 | 15–33 | 299 | 25.57 | 19–45 |

200 | 19.44 | 15–32 | 308 | 26.02 | 21–42 |

236 | 21.45 | 17–35 | 315 | 26.09 | 20–46 |

240 | 21.86 | 17–60 | 361 | 27.62 | 23–47 |

243 | 22.05 | 17–40 | 400 | 30.13 | 22–47 |

254 | 22.51 | 18–48 | 500 | 34.81 | 29–54 |

266 | 23.75 | 19–40 | 1000 | 51.80 | 43–70 |

101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 | 109 | 110 |

8 | 8 | 8 | 8 | 8 | 8 | 9 | 8 | 9 | 9 |

111 | 112 | 113 | 114 | 115 | 116 | 117 | 118 | 119 | 120 |

8 | 8 | 8 | 8 | 9 | 8 | 8 | 9 | 8 | 9 |

121–138 | 139–150 | 151–172 | 173–187 | 188–220 | / | / | / | / | / |

9 | 10 | 11 | 12 | 13 | / | / | / | / | / |

Order | Size | Parameter Set |
---|---|---|

197 | 13 | 4, 14, 17, 30, 35, 53, 54, 59, 77, 78, 87, 97, 98 |

200 | 13 | 14, 20, 45, 57, 67, 69, 70, 80, 82, 91, 92, 97, 99 |

236 | 14 | 2, 11, 24, 28, 51, 61, 69, 76, 81, 90, 103, 107, 110, 113 |

243 | 14 | 18, 20, 21, 24, 43, 47, 50, 52, 56, 59, 101, 105, 113, 116 |

254 | 15 | 11, 17, 31, 59, 67, 72, 74, 86, 92, 93, 100, 116, 119, 122, 125 |

266 | 15 | 3, 8, 19, 34, 47, 48, 54, 63, 69, 84, 89, 99, 112, 122, 124 |

290 | 17 | 12, 22, 32, 33, 56, 63, 71, 73, 79, 81, 82, 84, 86, 92, 100, 107, 121 |

299 | 18 | 12, 16, 19, 37, 45, 76, 81, 83, 89, 94, 96, 98, 119, 123, 130, 144, 145, 148 |

308 | 18 | 8, 9, 14, 26, 31, 37, 42, 43, 47, 49, 64, 67, 87, 108, 112, 127, 142, 146 |

315 | 18 | 26, 54, 74, 79, 89, 104, 119, 120, 127, 131, 135, 136, 138, 144, 149, 150, 155, 156 |

361 | 20 | 3, 9, 13, 17, 23, 24, 28, 29, 82, 83, 101, 102, 103, 141, 145, 151, 152, 153, 163, 171 |

400 | 21 | 18, 21, 24, 35, 37, 49, 52, 62, 64, 100, 108, 110, 115, 119, 130, 141, 142, 186, 189, 198, 199 |

Order | $\mathit{\alpha}\left(\mathit{G}\right)$ | Parameter Set |
---|---|---|

200 | 31 | 5, 13, 19, 20, 23, 35, 47, 49, 50, 59, 61, 76, 83, 86, 93 |

236 | 34 | 19, 40, 43, 49, 57, 61, 65, 67, 72, 75, 77, 78, 88, 90, 95, 111, 113 |

243 | 35 | 26, 37, 42, 43, 47, 50, 71, 78, 88, 96, 102, 107, 109, 111, 116, 117, 119 |

254 | 36 | 6, 9, 20, 21, 31, 47, 54, 57, 70, 82, 87, 89, 99, 112, 116, 123, 126 |

266 | 37 | 3, 11, 12, 28, 30, 32, 45, 51, 65, 67, 72, 82, 89, 98, 106, 108, 125, 131 |

290 | 40 | 6, 9, 26, 30, 34, 41, 42, 44, 46, 54, 57, 59, 61, 104, 117, 133, 135, 136, 137 |

299 | 41 | 11, 17, 18, 20, 23, 26, 51, 53, 61, 65, 66, 67, 75, 94, 96, 97, 100, 110, 125, 129 |

308 | 42 | 4, 9, 15, 20, 22, 28, 51, 62, 63, 76, 101, 103, 108, 109, 115, 120, 132, 134, 150, 153 |

315 | 43 | 1, 7, 9, 17, 29, 41, 45, 47, 53, 65, 73, 78, 84, 89, 104, 116, 128, 139, 141, 144, 155 |

361 | 47 | 3, 27, 36, 41, 64, 74, 84, 89, 95, 96, 112, 118, 127, 129, 134, 140, 142, 144, 146, 152, 162, 164, 166 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jiang, Y.; Liang, M.; Teng, Y.; Xu, X.
The Cyclic Triangle-Free Process. *Symmetry* **2019**, *11*, 955.
https://doi.org/10.3390/sym11080955

**AMA Style**

Jiang Y, Liang M, Teng Y, Xu X.
The Cyclic Triangle-Free Process. *Symmetry*. 2019; 11(8):955.
https://doi.org/10.3390/sym11080955

**Chicago/Turabian Style**

Jiang, Yu, Meilian Liang, Yanmei Teng, and Xiaodong Xu.
2019. "The Cyclic Triangle-Free Process" *Symmetry* 11, no. 8: 955.
https://doi.org/10.3390/sym11080955