Omidi, G. R.; Shahsiah, M. Diagonal Ramsey numbers of loose cycles in uniform hypergraphs. (English) Zbl 1368.05108 SIAM J. Discrete Math. 31, No. 3, 1634-1669 (2017). Summary: A \(k\)-uniform loose cycle \(\mathcal{C}_n^k\) is a hypergraph with vertex set \(\{v_1,v_2,\dots,v_{n(k-1)}\}\) and the set of \(n\) edges \(e_i=\{v_{(i-1)(k-1)+1},v_{(i-1)(k-1)+2},\dots, v_{(i-1)(k-1)+k}\}\), \(1\leq i\leq n\), where we use mod \(n(k-1)\) arithmetic. The diagonal Ramsey number of \(\mathcal{C}^k_n\), \(R(\mathcal{C}^k_n,\mathcal{C}^k_n)\), is asymptotically \(\frac{1}{2}(2k-1)n\), as has been proved by A. Gyárfás et al. [Electron. J. Comb. 15, No. 1, Research Paper R126, 14 p. (2008; Zbl 1165.05334)]. In this paper, we investigate to determine the exact value of \(R(\mathcal{C}^k_n,\mathcal{C}^k_n)\) and we show that for \(n\geq 2\) and \(k\geq 8,\) \(R(\mathcal{C}^k_n,\mathcal{C}^k_n)=(k-1)n+\lfloor\frac{n-1}{2}\rfloor.\) Cited in 1 ReviewCited in 4 Documents MSC: 05C65 Hypergraphs 05C55 Generalized Ramsey theory 05D10 Ramsey theory 05C38 Paths and cycles Keywords:Ramsey number; uniform hypergraph; loose path; loose cycle Citations:Zbl 1165.05334 PDFBibTeX XMLCite \textit{G. R. Omidi} and \textit{M. Shahsiah}, SIAM J. Discrete Math. 31, No. 3, 1634--1669 (2017; Zbl 1368.05108) Full Text: DOI arXiv References: [1] J. A. Bondy and P. Erdós, {\it Ramsey numbers for cycles in graphs}, J. Combin. Theory, Ser. B, 14 (1973), pp. 46-54. · Zbl 0248.05127 [2] D. Conlon, J. Fox, and B. Sudakov, {\it Ramsey numbers of sparse hypergraphs}, Random Structures Algorithms, 35 (2009), pp. 1-14. · Zbl 1203.05099 [3] O. Cooley, N. Fountoulakis, D. Kühn, and D. Osthus, 3-{\it uniform hypergraphs of bounded degree have linear Ramsey numbers}, J. Combin. Theory, Ser. B, 98 (2008), pp. 484-505. · Zbl 1140.05045 [4] O. Cooley, N. Fountoulakis, D. Kühn, and D. Osthus, {\it Embeddings and Ramsey numbers of sparse \(k\)-uniform hypergraphs}, Combinatorica, 29 (2009), pp. 263-297. · Zbl 1212.05170 [5] R. J. Faudree, S. L. Lawrence, T. D. Parsons, and R. H. Schelp, {\it Path-cycle Ramsey numbers}, Discrete Math., 10 (1974), pp. 269-277. · Zbl 0293.05120 [6] R. J. Faudree and R. H. Schelp, {\it All Ramsey numbers for cycles in graphs}, Discrete Math., 8 (1974), pp. 313-329. · Zbl 0294.05122 [7] A. Figaj and T. Ł uczak, {\it The Ramsey number for a triple of long even cycles}, J. Combin. Theory, Ser. B, 97 (2007), pp. 584-596. · Zbl 1120.05060 [8] L. Gerencsér and A. Gyárfás, {\it On Ramsey-type problems}, Ann. Univ. Scio. Budapest Eötvös Sect. Math., 10 (1967), pp. 167-170. · Zbl 0163.45502 [9] A. Gyárfás and G. Raeisi, {\it The Ramsey number of loose triangles and quadrangles in hypergraphs}, Electron. J. Combin., 19 (2012), #R30. · Zbl 1243.05161 [10] A. Gyárfás, M. Ruszinkó, G. Sárközy, and E. Szemerédi, {\it Three-color Ramsey numbers for paths}, Combinatorica, 27 (2007), pp. 35-69. · Zbl 1175.05093 [11] A. Gyárfás, G. Sárközy, and E. Szemerédi, {\it The Ramsey number of diamond-matchings and loose cycles in hypergraphs}, Electron. J. Combin., 15 (2008), #R126. · Zbl 1165.05334 [12] P. Haxell, T. Łuczak, Y. Peng, V. Rödl, A. Ruciński, M. Simonovits, and J. Skokan, {\it The Ramsey number for hypergraph cycles} I, J. Combin. Theory, Ser. A, 113 (2006), pp. 67-83. · Zbl 1089.05053 [13] L. Maherani, G. R. Omidi, G. Raeisi, and M. Shahsiah, {\it The Ramsey number of loose paths in 3-uniform hypergraphs}, Electron. J. Combin., 20 (2013), #P12. · Zbl 1266.05092 [14] B. Nagle, S. Olsen, V. Rödl, and M. Schacht, {\it On the Ramsey number of sparse 3-graphs}, Graphs Combin., 24 (2008), pp. 205-228. · Zbl 1169.05024 [15] G. R. Omidi and M. Shahsiah, {\it Ramsey numbers of 3-uniform loose paths and loose cycles}, J. Combin. Theory Ser. A, 121 (2014), pp. 64-73. · Zbl 1279.05051 [16] G. R. Omidi and M. Shahsiah, {\it Ramsey numbers of uniform loose paths and cycles}, Discrete Math., 340 (2017), pp. 1426-1434. · Zbl 1369.05142 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.