### The Ramsey Number of Loose Triangles and Quadrangles in Hypergraphs

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

#### Keywords

Hypergraph Ramsey Number, Loose Cycle, Loose Path