Regularity of symbolic powers of square-free monomial ideals

We study the regularity of symbolic powers of square-free monomial ideals. We prove that if I=I-Delta is the Stanley-Reisner ideal of a simplicial complex Delta, then reg(I-(n))<=delta(n-1)+ b for all n >= 1, where delta = lim(n) > infinity reg(I-(n))/n, b=max{reg(I-Gamma)vertical bar Gamma is a subcomplex of Delta with F(Gamma) subset of F(Delta)}, and F(Gamma) and F(Delta) are the set of facets of Gamma and Delta, respectively. This bound is sharp for any n. When I=I(G) is the edge ideal of a simple graph G, we obtain a general linear upper bound reg(I-(n)) <= 2n+ord-match(G) - 1, where ord-match( G) is the ordered matching number of G.