A question of Erdős and Selfridge [ErSe67], who could only show that $v_0(n)\geq 2$ for $n\geq 17$. More generally, they conjecture that \[v_l(n)=\max_{k\geq l}v(n,k)\to \infty\] as $n\to \infty$, for every fixed $l$, but could not even prove that $v_1(n)\geq 2$ for all large $n$.