Gyárfás–Györi–Simonovits partition conjecture for 3-uniform hypergraphs

Let $H$ be a $3$-uniform hypergraph, and let $b1(H)$ denote the cardinality of a largest independent set of $H$. A linear cycle is a sequence of at least three edges in which only cyclically consecutive edges intersect, and consecutive edges intersect in exactly one vertex. Gyárfás–Györi–Simonovits conjecture. The vertex set of every $3$-uniform hypergraph $H$ can be partitioned into $b1(H)$ linear cycles, edges, and subsets of hyperedges. This conjecture concerns decomposing the vertices of a hypergraph into pieces controlled by its independence number and linear-cycle structure; the source states that it remains open.