Gyárfás–Lehel–Sárközy–Szemerédi conjecture on monochromatic Hamiltonian Berge-cycles

Let $r\geq 2$ be fixed. An $r$-uniform hypergraph $K_n^r$ is the complete hypergraph on $n$ vertices. A Hamiltonian Berge-cycle is a Berge-cycle containing all $n$ vertices. An $k$-edge coloring assigns one of $k$ colors to every edge.

Gyárfás–Lehel–Sárközy–Szemerédi conjecture. For sufficiently large $n$, every $(r-1)$-edge coloring of $K_n^r$ contains a monochromatic Hamiltonian Berge-cycle.

Equivalently, for a given $r\geq 2$, the Ramsey number satisfies $R_{r-1}(C_n^{(r,2)})=n$ for sufficiently large $n$. The paper states this as the previously proposed conjecture and later proves the first open case $r=4$; the general assertion is therefore not established by the source.