Colored complete hypergraphs containing no rainbow Berge triangles

The study of graph Ramsey numbers within restricted colorings, in particular forbidding a rainbow triangle, has recently been blossoming under the name Gallai-Ramsey numbers. In this work, we extend the main structural tool from rainbow triangle free colorings of complete graphs to rainbow Berge triangle free colorings of hypergraphs. In doing so, some other concepts and results are also translated from graphs to hypergraphs

Monochromatic kk-connected Subgraphs in 2-edge-colored Complete Graphs

Bollob\'{a}s and Gy\'{a}rf\'{a}s conjectured that for any $k, n \in \mathbb{Z}^+$ with $n > 4(k-1)$, every 2-edge-coloring of the complete graph on $n$ vertices leads to a $k$-connected monochromatic subgraph with at least $n-2k+2$ vertices. We find a counterexample with $n = 5k-2\lceil\sqrt{2k-1}\rceil-3$, thus disproving the conjecture, and we show the conjecture is true for $n \ge 5k-\min\{\sqrt{4k-2}+3, 0.5k+4\}$