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Diagonal Ramsey numbers of loose cycles in uniform hypergraphs. (English) Zbl 1368.05108

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 \(k\geq 8,\) \(R(\mathcal{C}^k_n,\mathcal{C}^k_n)=(k-1)n+\lfloor\frac{n-1}{2}\rfloor.\)

MSC:

05C65 Hypergraphs
05C55 Generalized Ramsey theory
05D10 Ramsey theory
05C38 Paths and cycles

Citations:

Zbl 1165.05334
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References:

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