Polynomial bounds for monochromatic tight cycle partition in r-edge-coloured K^(k)_n

Let K^(k)_n be the complete k-graph on n vertices. A k-uniform tight cycle is a k-graph with its vertices cyclically ordered so that every k consecutive vertices form an edge and any two consecutive edges share exactly k−1 vertices. A result of Bustamante, Corsten, Frankl, Pokrovskiy and Skokan shows that all r-edge coloured K^(k)_n can be partitioned into c_r,k vertex disjoint monochromatic tight cycles. However, the constant c_r,k is of tower-type. In this work, we show that c_r,k is a polynomial in r.