Abstract: In 1999, Łuczak proved that the three-coloured Ramsey number of the cycle of length $n$ is less than $(4+o(1))n$, presenting a technique that, conceptually, through the use of Szemerédi’s regularity lemma, reduces the problem to that of finding the Ramsey number of a connected matching. Ever since then, Łuczak’s method has been applied successfully in many results.
There are many natural ways to generalize cycles for $k$-uniform hypergraphs. In this talk, we first present a brief survey of the Ramsey numbers of Berge, loose, and tight cycles. Then we examine the use of Łuczak’s connected matching method on hypergraphs, with a focus on the loose cycle case.