The dataset viewer is not available for this dataset.
Error code: ConfigNamesError
Exception: RuntimeError
Message: Dataset scripts are no longer supported, but found RamseyGraph.py
Traceback: Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/dataset/config_names.py", line 66, in compute_config_names_response
config_names = get_dataset_config_names(
^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/inspect.py", line 161, in get_dataset_config_names
dataset_module = dataset_module_factory(
^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/load.py", line 1032, in dataset_module_factory
raise e1 from None
File "/usr/local/lib/python3.12/site-packages/datasets/load.py", line 992, in dataset_module_factory
raise RuntimeError(f"Dataset scripts are no longer supported, but found {filename}")
RuntimeError: Dataset scripts are no longer supported, but found RamseyGraph.pyNeed help to make the dataset viewer work? Make sure to review how to configure the dataset viewer, and open a discussion for direct support.
| A 6-vertex Ramsey(4, 4) graph and its complement |
|---|
 |
| It contains no complete subgraph of 4 vertices, nor does it contain an independent set of 4 vertices. |
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.
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.
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!
For the latest research on Ramsey graphs, please refer to Radziszowski's dynamic survey, continuously updated in the Electronic Journal of Combinatorics.
Ramsey Numbers
Ramsey numbers refer to the minimum number of vertices satisfying Ramsey graphs. Here are some known Ramsey numbers:
| s\t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | - | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 3 | - | - | 6 | 9 | 14 | 18 | 23 | 28 | 36 | 40 - 41 |
| 4 | - | - | - | 18 | 25 | 36 - 40 | 49 - 58 | 59 - 79 | 73 - 106 | 92 - 136 |
| 5 | - | - | - | - | 43 - 48 | 59 - 85 | 80 - 133 | 101 - 194 | 133 - 282 | 149 - 381 |
| 6 | - | - | - | - | - | 102 - 161 | 115 - 273 | 134 - 427 | 183 - 656 | 204 - 949 |
| 7 | - | - | - | - | - | - | 205 - 497 | 219 - 840 | 252 - 1379 | 292 - 2134 |
| 8 | - | - | - | - | - | - | - | 282 - 1532 | 329 - 2683 | 343 - 4432 |
| 9 | - | - | - | - | - | - | - | - | 565 - 5366 | 581 - 9797 |
| 10 | - | - | - | - | - | - | - | - | - | 798 - 17730 |
How to Use
This repository hosts graphs related to the classical Ramsey Numbers. You can access them using the following code:
pip install datasets
>>> from datasets import load_dataset
>>> dataset = load_dataset("linxy/RamseyGraph", "r44_3", trust_remote_code=True)
>>> for i in dataset["train"]:
>>> print(i)
{'edges': [], 'num_nodes': 3}
{'edges': [[1, 2]], 'num_nodes': 3}
{'edges': [[0, 2], [1, 2]], 'num_nodes': 3}
{'edges': [[0, 1], [0, 2], [1, 2]], 'num_nodes': 3}
r44_3 refers to Ramsey(4,4) graphs with 3 vertices. Here are all dataset names:
['r34_1', 'r34_2', 'r34_3', 'r34_4', 'r34_5', 'r34_6', 'r34_7', 'r34_8',
'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',
'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',
'r37_21', 'r37_22',
'r38_27',
'r39_35',
'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',
'r45_24',
'r46_35some',
'r55_42some']
Progress
Many Ramsey graphs have been found, but many remain undiscovered. Here are some known Ramsey graphs (also available in data/ directory):
- All maximal Ramsey(3,7) graphs
- 21 vertices (compressed) (1118436 graphs, discovered by Gunnar Brinkmann and Jan Goedgebeur)
- 22 vertices (191 graphs)
- All maximal Ramsey(3,8) graphs
- 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.
- 27 vertices (compressed) (477142 graphs)
- All maximal Ramsey(3,9) graphs
- 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 by Goedgebeur and Radziszowski.
- 35 vertices (1 graph)
- All maximal Ramsey(4,5) graphs
- 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.
- Known largest Ramsey(4,6) graphs
- 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.
- Known largest Ramsey(5,5) graphs
- 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 contains 328 of these graphs, with the other 328 being their complements.
- Ramsey(4,4;3)-hypergraphs
- 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.
Acknowledgments
The ramsey database by Gunnar Brinkmann and Jan Goedgebeur
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