Erdős–Gyárfás monochromatic-same-color path cover conjecture

Let $K_n$ be a complete graph whose edges are colored red and blue. A same-color monochromatic path cover is a collection of monochromatic paths, all in one common color, whose vertices cover $V(K_n)$.

Erdős–Gyárfás same-color path cover conjecture. The vertex set of every $2$-colored $K_n$ can be covered by at most

$$\sqrt{n}$$

monochromatic paths of the same color. This would be best possible.

A bound of $2\sqrt n$ is known. The sharper bound remains open.