Gyárfás–Sárközy's monochromatic loose path partition conjecture

Let $K_n^k$ be the complete $k$-uniform hypergraph on $n$ vertices, with its edges colored using two colors. A loose path is a hypergraph path in which consecutive edges intersect in exactly one vertex and nonconsecutive edges are disjoint; sets of fewer than $k$ vertices are also accepted as loose paths of either color. Gyárfás–Sárközy's conjecture. There are two disjoint monochromatic loose paths of distinct colors covering all but at most $k-2$ vertices. This estimate is sharp for sufficiently large $n$. The conjecture generalizes the two-color path-partition result for complete graphs to loose paths in uniform hypergraphs; the stated sharpness indicates that the uncovered-vertex bound cannot generally be improved.