Grinshpun–Sárközy linear bound for powers-of-cycles covers

Let $K$ be a complete graph whose edges are colored red and blue, and let the $k$-th power of a cycle be the graph obtained by joining vertices whose cyclic distance is at most $k$. A monochromatic cover consists of vertex-disjoint monochromatic copies of this graph whose vertices cover $V(K)$.

Grinshpun–Sárközy powers-of-cycles conjecture. The vertex set of every $2$-colored complete graph can be covered by at most

$$ck$$

vertex-disjoint monochromatic $k$-th powers of a cycle, for an absolute constant $c$.

The known bound is exponential in $k\log k$; the conjectured linear dependence on $k$ remains open.