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🎉 [update] Improve README formatting and add badges
Browse files- README.md +65 -56
- README_CN.md +132 -0
README.md
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---
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# Ramsey Graph
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```bash
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pip install datasets
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{'edges': [[0, 1], [0, 2], [1, 2]], 'num_nodes': 3}
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```
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`r44_3`
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```python
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['r34_1', 'r34_2', 'r34_3', 'r34_4', 'r34_5', 'r34_6', 'r34_7', 'r34_8',
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'r35_1', 'r35_2', 'r35_3', 'r35_4', 'r35_5', 'r35_6', 'r35_7', 'r35_8', 'r35_9', 'r35_10', 'r35_11', 'r35_12', 'r35_13',
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'r55_42some']
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```
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#
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| --------------- |
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| ](https://opensource.org/licenses/MIT)
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[](https://huggingface.co/datasets/linxy/RamseyGraph)
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[](https://linxueyuan.online/RamseyGraph)
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[](https://huggingface.co/datasets/linxy/RamseyGraph)
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[🌐 Live Demo](https://linxueyuan.online/RamseyGraph) | [中文文档](README_CN.md)
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</div>
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This repository hosts graphs related to the classical **Ramsey Numbers**. You can access them using the following code:
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```bash
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pip install datasets
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{'edges': [[0, 1], [0, 2], [1, 2]], 'num_nodes': 3}
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```
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`r44_3` refers to Ramsey(4,4) graphs with 3 vertices. Here are all dataset names:
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```python
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['r34_1', 'r34_2', 'r34_3', 'r34_4', 'r34_5', 'r34_6', 'r34_7', 'r34_8',
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'r35_1', 'r35_2', 'r35_3', 'r35_4', 'r35_5', 'r35_6', 'r35_7', 'r35_8', 'r35_9', 'r35_10', 'r35_11', 'r35_12', 'r35_13',
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'r55_42some']
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```
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# Introduction
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| A 6-vertex Ramsey(4, 4) graph and its complement |
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| --------------- |
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| It contains no complete subgraph of 4 vertices, nor does it contain an independent set of 4 vertices. |
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A **Ramsey(s,t,n) graph** is a graph with $n$ vertices that contains no clique of size $s$ and no independent set of size $t$. The `n` is usually omitted, and **Ramsey(s,t) graph** is used to refer to Ramsey(s,t,n) graphs for some $n$. **Ramsey's theorem** states that for given $s$ and $t$, the number of Ramsey(s,t) graphs is finite. The minimum number of vertices satisfying a Ramsey graph is called a **Ramsey Number**. However, finding all such graphs, or even determining the maximum $n$ for which they exist, is a famous combinatorial mathematics problem.
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The Ramsey numbers known to humans are very limited, with most only having known upper and lower bounds. One approach is to search for the largest Ramsey graphs, whose number of vertices provides a lower bound for the Ramsey number.
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If you are interested in this topic, you can try to find the **largest Ramsey(5,5) graph**. All Ramsey(5,5) graphs with 42 vertices have been found, but it is uncertain whether there are any with 43 vertices. The lower bound for the Ramsey number Ramsey(5,5) was last improved in 1989. If you find even one such graph, it would be a significant advancement in this field after 35 years!
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> For the latest research on Ramsey graphs, please refer to **Radziszowski**'s dynamic survey, continuously updated in the [**Electronic Journal of Combinatorics**](https://www.combinatorics.org).
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## Ramsey Numbers
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**Ramsey numbers** refer to the minimum number of vertices satisfying Ramsey graphs. Here are some known Ramsey numbers:
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| s\t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
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| --- | --- | --- | --- | --- | ------- | --------- | --------- | ---------- | ---------- | ----------- |
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| 9 | - | - | - | - | - | - | - | - | 565 - 5366 | 581 - 9797 |
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| 10 | - | - | - | - | - | - | - | - | - | 798 - 17730 |
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## Progress
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Many Ramsey graphs have been found, but many remain undiscovered. Here are some known Ramsey graphs:
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| Vertices | Ramsey(3,4) | Ramsey(3,5) | Ramsey(3,6) | Ramsey(4,4) Graphs |
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| ------ | ------------------------ | -------------------------- | ------------------------------------------- | -------------------------------------------- |
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| 1 | [1 graph](data/r34_1.g6) | [1 graph](data/r35_1.g6) | [1 graph](data/r36_1.g6) | [1 graph](data/r44_1.g6) |
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| 2 | [2 graphs](data/r34_2.g6) | [2 graphs](data/r35_2.g6) | [2 graphs](data/r36_2.g6) | [2 graphs](data/r44_2.g6) |
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| 3 | [3 graphs](data/r34_3.g6) | [3 graphs](data/r35_3.g6) | [3 graphs](data/r36_3.g6) | [4 graphs](data/r44_3.g6) |
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| 4 | [6 graphs](data/r34_4.g6) | [7 graphs](data/r35_4.g6) | [7 graphs](data/r36_4.g6) | [9 graphs](data/r44_4.g6) |
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| 5 | [9 graphs](data/r34_5.g6) | [13 graphs](data/r35_5.g6) | [14 graphs](data/r36_5.g6) | [24 graphs](data/r44_5.g6) |
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| 6 | [15 graphs](data/r34_6.g6) | [32 graphs](data/r35_6.g6) | [37 graphs](data/r36_6.g6) | [84 graphs](data/r44_6.g6) |
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| 7 | [9 graphs](data/r34_7.g6) | [71 graphs](data/r35_7.g6) | [100 graphs](data/r36_7.g6) | [362 graphs](data/r44_7.g6) |
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| 8 | [3 graphs](data/r34_8.g6) | [179 graphs](data/r35_8.g6) | [356 graphs](data/r36_8.g6) | [2079 graphs](data/r44_8.g6) |
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| 9 | | [290 graphs](data/r35_9.g6) | [1407 graphs](data/r36_9.g6) | [14701 graphs](data/r44_9.g6) |
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| 10 | | [313 graphs](data/r35_10.g6) | [6657 graphs](data/r36_10.g6) | [103706 graphs (compressed)](data/r44_10.g6.gz) |
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| 11 | | [105 graphs](data/r35_11.g6) | [30395 graphs (compressed)](data/r36_11.g6.gz) | [546356 graphs (compressed)](data/r44_11.g6.gz) |
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| 12 | | [12 graphs](data/r35_12.g6) | [116792 graphs (compressed)](data/r36_12.g6.gz) | [1449166 graphs (compressed)](data/r44_12.g6.gz) |
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| 13 | | [1 graph](data/r35_13.g6) | [275086 graphs (compressed)](data/r36_13.g6.gz) | [1184231 graphs (compressed)](data/r44_13.g6.gz) |
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| 14 | | | [263520 graphs (compressed)](data/r36_14.g6.gz) | [130816 graphs (compressed)](data/r44_14.g6.gz) |
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| 15 | | | [64732 graphs (compressed)](data/r36_15.g6.gz) | [640 graphs](data/r44_15.g6) |
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| 16 | | | [2576 graphs (compressed)](data/r36_16.g6.gz) | [2 graphs](data/r44_16.g6) |
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| 17 | | | [7 graphs](data/r36_17.g6) | [1 graph](data/r44_17.g6) |
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- All maximal Ramsey(3,7) graphs
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- [21 vertices (compressed)](data/r37_21.g6.gz) (1118436 graphs, discovered by **Gunnar Brinkmann** and **Jan Goedgebeur**)
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- [22 vertices](data/r37_22.g6) (191 graphs)
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- All maximal Ramsey(3,8) graphs
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- In 1992, **McKay** and **Zhang** proved that the maximal Ramsey(3,8) graph has 27 vertices, but the complete set of Ramsey(3,8,27) graphs was not determined until 2012 by **Gunnar Brinkmann** and **Jan Goedgebeur**.
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- [27 vertices (compressed)](data/r38_27.g6.gz) (477142 graphs)
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- All maximal Ramsey(3,9) graphs
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- The maximal Ramsey(3,9) graph has 35 vertices, discovered long ago by **Kalbfleisch**, but its uniqueness was not proven until 2013. See the [paper](https://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i1p30) by **Goedgebeur** and **Radziszowski**.
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- [35 vertices](data/r39_35.g6) (1 graph)
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- All maximal Ramsey(4,5) graphs
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- In 1995, **McKay** and **Radziszowski** proved that there are no Ramsey(4,5) graphs with more than 24 vertices and found 350904 graphs with 24 vertices. The remaining graphs were discovered in 2016 by **McKay** and **Angeltveit**. There are 352366 graphs in total, see [r45_24.g6](data/r45_24.g6).
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- Known largest Ramsey(4,6) graphs
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- In early 2012, **Geoffrey Exoo** discovered 37 Ramsey(4,6,35) graphs. There may be more, and graphs with 36 to 40 vertices may even exist. See [r46_35some.g6](data/r46_35some.g6).
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- Known largest Ramsey(5,5) graphs
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- In 1989, **Geoffrey Exoo** discovered several Ramsey(5,5,42) graphs. **McKay** and **Radziszowski** extended this to 656 graphs and conjectured that larger graphs are impossible. However, there may be more 42-vertex graphs, and graphs with 43 to 47 vertices may even exist. [r55_42some.g6](data/r55_42some.g6) contains 328 of these graphs, with the other 328 being their complements.
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- Ramsey(4,4;3)-hypergraphs
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- A **Ramsey(4,4;3) hypergraph** is a 3-uniform hypergraph that cannot contain a complete subgraph with 4 vertices, nor a complete independent set with 4 vertices. **Steve Butler** and **Aaron Wootton** discovered 42 such hypergraphs in 2010, each with 13 vertices.
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## Acknowledgments
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The [**ramsey**](https://users.cecs.anu.edu.au/~bdm/data/ramsey.html) database by **Gunnar Brinkmann** and **Jan Goedgebeur**
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README_CN.md
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---
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license: mit
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task_categories:
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- graph-ml
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language:
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- zh
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size_categories:
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- 100M<n<1B
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---
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# Ramsey Graph
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<div align="center">
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[](https://opensource.org/licenses/MIT)
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[](https://huggingface.co/datasets/linxy/RamseyGraph)
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[](https://linxueyuan.online/RamseyGraph)
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[](https://huggingface.co/datasets/linxy/RamseyGraph)
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[🌐 在线演示](https://linxueyuan.online/RamseyGraph)
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</div>
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本仓库托管了一些与经典 **拉姆齐数（Ramsey Number）** 相关的图。你可以使用以下代码获取：
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```bash
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pip install datasets
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```
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```python
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>>> from datasets import load_dataset
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>>> dataset = load_dataset("linxy/RamseyGraph", "r44_3", trust_remote_code=True)
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>>> for i in dataset["train"]:
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>>> print(i)
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{'edges': [], 'num_nodes': 3}
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{'edges': [[1, 2]], 'num_nodes': 3}
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{'edges': [[0, 2], [1, 2]], 'num_nodes': 3}
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{'edges': [[0, 1], [0, 2], [1, 2]], 'num_nodes': 3}
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```
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`r44_3` 指 Ramsey(4,4) 图中 3 个顶点的图。以下是所有数据集的名称：
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```python
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['r34_1', 'r34_2', 'r34_3', 'r34_4', 'r34_5', 'r34_6', 'r34_7', 'r34_8',
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'r35_1', 'r35_2', 'r35_3', 'r35_4', 'r35_5', 'r35_6', 'r35_7', 'r35_8', 'r35_9', 'r35_10', 'r35_11', 'r35_12', 'r35_13',
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'r36_1', 'r36_2', 'r36_3', 'r36_4', 'r36_5', 'r36_6', 'r36_7', 'r36_8', 'r36_9', 'r36_10', 'r36_11', 'r36_12', 'r36_13', 'r36_14', 'r36_15', 'r36_16', 'r36_17',
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'r37_21', 'r37_22',
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'r38_27',
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'r39_35',
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'r44_1', 'r44_2', 'r44_3', 'r44_4', 'r44_5', 'r44_6', 'r44_7', 'r44_8', 'r44_9', 'r44_10', 'r44_11', 'r44_12', 'r44_13', 'r44_14', 'r44_15', 'r44_16', 'r44_17',
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'r45_24',
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'r46_35some',
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'r55_42some']
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```
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# 介绍
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| 一个 6 结点的 Ramsey(4, 4) 图及其补图 |
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| --------------- |
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| 它不包含 4 个顶点的完全子图，也不包含 4 个顶点的完全独立集。 |
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**Ramsey(s,t,n) 图** 是具有 $n$ 个顶点的图,它不包含大小为 $s$ 的团，也不包含大小为 $t$ 的独立集。通常将 `n` 省略，用 **Ramsey(s,t) 图** 代指某些 $n$ 的 Ramsey(s,t,n) 图。**拉姆齐定理**表示，对于给定的 $s$ 和 $t$，Ramsey(s,t) 图的数量是有限的。我们称满足 Ramsey 图的最小顶点数为**拉姆齐数（Ramsey Number）**。然而，找到所有这样的图，甚至确定它们存在的最大 $n$，都是一个著名的组合数学难题。
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人类已知的拉姆齐数非常有限，大部分只能知道该数的上界和下界。一个方法是寻找最大的拉姆齐图，它的顶点数就是拉姆齐数的下界。
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如果你对这个主题感兴趣，可以尝试找一下 **最大的 Ramsey(5,5) 图**。人们已经将 42 顶点的 Ramsey(5,5) 图全部找到了，但是不确定有没有 43 顶点的 Ramsey(5,5) 图。拉姆齐数 Ramsey(5,5) 的下界最后一次被改进是在 1989 年。只要你找到一个，那就是这个领域 35 年来的重要进展！
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> 有关拉姆齐图的最新研究，请参见 **Radziszowski** 的动态综述，持续更新刊登于 [**电子组合学期刊**](https://www.combinatorics.org)。
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## 拉姆齐数
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**拉姆齐数** 是指满足拉姆齐图的最小顶点数。以下是一些已知的拉姆齐数：
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| s\t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
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| --- | --- | --- | --- | --- | ------- | --------- | --------- | ---------- | ---------- | ----------- |
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| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
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| 2 | - | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
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| 3 | - | - | 6 | 9 | 14 | 18 | 23 | 28 | 36 | 40 - 41 |
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| 4 | - | - | - | 18 | 25 | 36 - 40 | 49 - 58 | 59 - 79 | 73 - 106 | 92 - 136 |
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| 5 | - | - | - | - | 43 - 48 | 59 - 85 | 80 - 133 | 101 - 194 | 133 - 282 | 149 - 381 |
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| 81 |
+
| 6 | - | - | - | - | - | 102 - 161 | 115 - 273 | 134 - 427 | 183 - 656 | 204 - 949 |
|
| 82 |
+
| 7 | - | - | - | - | - | - | 205 - 497 | 219 - 840 | 252 - 1379 | 292 - 2134 |
|
| 83 |
+
| 8 | - | - | - | - | - | - | - | 282 - 1532 | 329 - 2683 | 343 - 4432 |
|
| 84 |
+
| 9 | - | - | - | - | - | - | - | - | 565 - 5366 | 581 - 9797 |
|
| 85 |
+
| 10 | - | - | - | - | - | - | - | - | - | 798 - 17730 |
|
| 86 |
+
|
| 87 |
+
## 进展
|
| 88 |
+
|
| 89 |
+
目前人们已经找到了许多拉姆齐图，但仍有许多图尚未找到。以下是一些已知的拉姆齐图：
|
| 90 |
+
|
| 91 |
+
| 顶点数 | Ramsey(3,4) | Ramsey(3,5) | Ramsey(3,6) | Ramsey(4,4) 图 |
|
| 92 |
+
| ------ | ------------------------ | -------------------------- | ------------------------------------------- | -------------------------------------------- |
|
| 93 |
+
| 1 | [1 个图](data/r34_1.g6) | [1 个图](data/r35_1.g6) | [1 个图](data/r36_1.g6) | [1 个图](data/r44_1.g6) |
|
| 94 |
+
| 2 | [2 个图](data/r34_2.g6) | [2 个图](data/r35_2.g6) | [2 个图](data/r36_2.g6) | [2 个图](data/r44_2.g6) |
|
| 95 |
+
| 3 | [3 个图](data/r34_3.g6) | [3 个图](data/r35_3.g6) | [3 个图](data/r36_3.g6) | [4 个图](data/r44_3.g6) |
|
| 96 |
+
| 4 | [6 个图](data/r34_4.g6) | [7 个图](data/r35_4.g6) | [7 个图](data/r36_4.g6) | [9 个图](data/r44_4.g6) |
|
| 97 |
+
| 5 | [9 个图](data/r34_5.g6) | [13 个图](data/r35_5.g6) | [14 个图](data/r36_5.g6) | [24 个图](data/r44_5.g6) |
|
| 98 |
+
| 6 | [15 个图](data/r34_6.g6) | [32 个图](data/r35_6.g6) | [37 个图](data/r36_6.g6) | [84 个图](data/r44_6.g6) |
|
| 99 |
+
| 7 | [9 个图](data/r34_7.g6) | [71 个图](data/r35_7.g6) | [100 个图](data/r36_7.g6) | [362 个图](data/r44_7.g6) |
|
| 100 |
+
| 8 | [3 个图](data/r34_8.g6) | [179 个图](data/r35_8.g6) | [356 个图](data/r36_8.g6) | [2079 个图](data/r44_8.g6) |
|
| 101 |
+
| 9 | | [290 个图](data/r35_9.g6) | [1407 个图](data/r36_9.g6) | [14701 个图](data/r44_9.g6) |
|
| 102 |
+
| 10 | | [313 个图](data/r35_10.g6) | [6657 个图](data/r36_10.g6) | [103706 个图 (压缩文件)](data/r44_10.g6.gz) |
|
| 103 |
+
| 11 | | [105 个图](data/r35_11.g6) | [30395 个图 (压缩文件)](data/r36_11.g6.gz) | [546356 个图 (压缩文件)](data/r44_11.g6.gz) |
|
| 104 |
+
| 12 | | [12 个图](data/r35_12.g6) | [116792 个图 (压缩文件)](data/r36_12.g6.gz) | [1449166 个图 (压缩文件)](data/r44_12.g6.gz) |
|
| 105 |
+
| 13 | | [1 个图](data/r35_13.g6) | [275086 个图 (压缩文件)](data/r36_13.g6.gz) | [1184231 个图 (压缩文件)](data/r44_13.g6.gz) |
|
| 106 |
+
| 14 | | | [263520 个图 (压缩文件)](data/r36_14.g6.gz) | [130816 个图 (压缩文件)](data/r44_14.g6.gz) |
|
| 107 |
+
| 15 | | | [64732 个图 (压缩文件)](data/r36_15.g6.gz) | [640 个图](data/r44_15.g6) |
|
| 108 |
+
| 16 | | | [2576 个图 (压缩文件)](data/r36_16.g6.gz) | [2 个图](data/r44_16.g6) |
|
| 109 |
+
| 17 | | | [7 个图](data/r36_17.g6) | [1 个图](data/r44_17.g6) |
|
| 110 |
+
|
| 111 |
+
- 所有最大 Ramsey(3,7) 图
|
| 112 |
+
- [21 个顶点 (压缩文件)](data/r37_21.g6.gz) (1118436 个图，由 **Gunnar Brinkmann** 和 **Jan Goedgebeur** 发现)
|
| 113 |
+
- [22 个顶点](data/r37_22.g6) (191 个图)
|
| 114 |
+
- 所有最大 Ramsey(3,8) 图
|
| 115 |
+
- 1992 年 **McKay** 和 **Zhang** 证明最大 Ramsey(3,8) 图有 27 个顶点，但完整的 Ramsey(3,8,27) 图集直到 2012 年才由 **Gunnar Brinkmann** 和 **Jan Goedgebeur** 确定。
|
| 116 |
+
- [27 个顶点 (压缩文件)](data/r38_27.g6.gz) (477142 个图)
|
| 117 |
+
- 所有最大 Ramsey(3,9) 图
|
| 118 |
+
- 最大 Ramsey(3,9) 图有 35 个顶点，由 **Kalbfleisch** 很久以前发现，但直到 2013 年才证明其唯一性。参见 **Goedgebeur** 和 **Radziszowski** 的[论文](https://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i1p30)。
|
| 119 |
+
- [35 个顶点](data/r39_35.g6) (1 个图)
|
| 120 |
+
- 所有最大 Ramsey(4,5) 图
|
| 121 |
+
- 1995 年，**McKay** 和 **Radziszowski** 证明不存在超过 24 个顶点的 Ramsey(4,5) 图，并找到了 350904 个 24 顶点的图。剩下的图在 2016 年由 **McKay** 和 **Angeltveit** 发现。总共有 352366 个图，参见 [r45_24.g6](data/r45_24.g6)。
|
| 122 |
+
- 已知最大的 Ramsey(4,6) 图
|
| 123 |
+
- 2012 年初，**Geoffrey Exoo** 发现了 37 个 Ramsey(4,6,35) 图。这可能还有更多，甚至可能存在 36 到 40 个顶点的图。参见 [r46_35some.g6](data/r46_35some.g6)。
|
| 124 |
+
- 已知最大的 Ramsey(5,5) 图
|
| 125 |
+
- 1989 年，**Geoffrey Exoo** 发现了几个 Ramsey(5,5,42) 图。**McKay** 和 **Radziszowski** 将其扩展至 656 个图，并推测不可能有更大的图。然而，可能还有更多 42 顶点的图，甚至可能存在 43 到 47 个顶点的图。[r55_42some.g6](data/r55_42some.g6) 包含其中 328 个图，其他 328 个是它们的补图。
|
| 126 |
+
- Ramsey(4,4;3)-超图
|
| 127 |
+
- **Ramsey(4,4;3) 超图** 是一个 3-均匀超图，不能包含 4-顶点的完全子图，也不能包含 4-顶点的完全独立集。**Steve Butler** 和 **Aaron Wootton** 在 2010 年发现了 42 个这样的超图，每个都有 13 个顶点。
|
| 128 |
+
|
| 129 |
+
|
| 130 |
+
## 致谢
|
| 131 |
+
|
| 132 |
+
**Gunnar Brinkmann** 和 **Jan Goedgebeur** 的 [**ramsey**](https://users.cecs.anu.edu.au/~bdm/data/ramsey.html) 数据库
|