The Ramsey number of loose triangles and quadrangles in hypergraphs (Q426909)

Summary: Asymptotic values of hypergraph Ramsey numbers for loose cycles (and paths) were determined recently. Here we determine some of them exactly, for example the 2-color hypergraph Ramsey number of a \(k\)-uniform loose 3-cycle or 4-cycle: \(R(\mathcal{C}^k_3,\mathcal{C}^k_3)=3k-2\) and \(R(\mathcal{C}_4^k,\mathcal{C}_4^k)=4k-3 (for k\geq 3)\). For more than 3-colors we could prove only that \(R(\mathcal{C}^3_3,\mathcal{C}^3_3,\mathcal{C}^3_3)=8\). Nevertheless, the \(r\)-color Ramsey number of triangles for hypergraphs are much smaller than for graphs: for \(r\geq 3\), \[ r+5\leq R(\mathcal{C}_3^3,\mathcal{C}_3^3,\dots,\mathcal{C}_3^3)\leq 3r \]