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A class of Ramsey-extremal hypergraphs | ||
| Transactions on Combinatorics | ||
| مقاله 4، دوره 6، شماره 3، آذر 2017، صفحه 37-43 اصل مقاله (214.26 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2017.21468 | ||
| نویسنده | ||
| Brendan D. McKay* | ||
| Australian National University | ||
| چکیده | ||
| In 1991, McKay and Radziszowski proved that, however each $3$-subset of a $13$-set is assigned one of two colours, there is some $4$-subset whose four $3$-subsets have the same colour. More than 25 years later, this remains the only non-trivial classical Ramsey number known for hypergraphs. In this article, we find all the extremal colourings of the $3$-subsets of a 12-set and list some of their properties. We also provide an answer to a question of Dudek, La Fleur, Mubayi and R\"odl about the size-Ramsey numbers of hypergraphs. | ||
| کلیدواژهها | ||
| hypergraph؛ Ramsey number؛ size-Ramsey number | ||
| مراجع | ||
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[1] A. Dudek, S. La Fleur, D. Mubayi and V. Rodl, On the size-Ramsey numb er of hyp ergraphs, J. Graph Theory, to app ear, 2015, https://arxiv.org/abs/1503.06304.
[2] J. R. Isb ell, R (4;4;3)≥13, J. Combin. Th., 6 (1969).
[3] B. D. McKay, Combinatorial data, Online at, http://users.cecs.anu.edu.au/ b dm/data/ramsey.html.
[4] B. D. McKay, Isomorph-free exhaustive generation, J. Algorithms, 26 (1998) 306-324.
[5] B. D. McKay and A. Pip erno, Practical Graph Isomorphism, II, J. Symbolic Comput., 60 (2014) 94-112.
[6] B. D. McKay and S. P. Radziszowski, The rst classical Ramsey number for hypergraphs is computed, Pro ceedings of the Second Annual ACM-SIAM Symp osium on Discrete Algorithms, SODA'91, San Francisco, (1991) 304-308.
[7] B. D. McKay and S. P. Radziszowski, R(4;5)=25, J. Graph Theory, 19 (1995) 309-322.
[8] S. P. Radziszowski, Small Ramsey Numb ers, Electron. J. Combin., 1 (1994) pp. 30. | ||
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