SOLVED

Is it true that, in any two-colouring of the edges of $K_n$, there exist $sqrt{n}$ monochromatic paths, all of the same colour, which cover all vertices?

A problem of Erdős and Gyárfás. Gerencsér and Gyárfás [GeGy67] proved that, if the paths do not need to be of the same colour, then two paths suffice. Erdős and Gyárfás [ErGy95] proved that $2\sqrt{n}$ vertices suffice, and observed that $\sqrt{n}$ would be best possible here.

Solved in the affirmative by Pokrovskiy, Versteegen, and Williams [PVW24].