On the Waldschmidt constant of square-free principal Borel ideals

Fix a square-free monomial $m in S = mathbb{K}[x_1,ldots,x_n]$. The square-free principal Borel ideal generated by $m$, denoted ${rm sfBorel}(m)$, is the ideal generated by all the square-free monomials that can be obtained via Borel moves from the monomial $m$. We give upper and lower bounds for the Waldschmidt constant of ${rm sfBorel}(m)$ in terms of the support of $m$, and in some cases, exact values. For any rational $frac{a}{b} geq 1$, we show that there exists a square-free principal Borel ideal with Waldschmidt constant equal to $frac{a}{b}$.