The Ramsey Number of Loose Triangles and Quadrangles in Hypergraphs

Source abstract

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 $r\geq 3$, $$r+5\le R(\mathcal{C}_3^3,\mathcal{C}_3^3,\dots,\mathcal{C}_3^3)\le 3r$$