A linear gap for the maximal cp-rank

Abstract

We prove that the maximal cp-rank $p_n$ satisfies $p_n\ge n(n-3)/2$ for odd $n\ge 5$ and $p_n\ge n(n-3)/2-1$ for even $n\ge 6$. Together with the known upper bound $p_n\le \left(\genfrac{}{}{0ex}{}{n+1}{2}\right)-4$, this gives $p_n=n^2/2+O(n)$.