Abstract. Given a squarefree monomial ideal I ⊆ R = k[x 1 , . . . , x n ], we show that α(I), the Waldschmidt constant of I, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of I. By applying results from fractional graph theory, we can then express α(I) in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of I. Moreover, expressing α(I) as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on α(I), thus verifying a conjecture of Cooper-Embree-Hà-Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals. For example, we determine both constants for unions of general linear subspaces of P n with few components compared to n, and we find the Waldschmidt constant for the Stanley-Reisner ideal of a uniform matroid.

Given a nontrivial homogeneous ideal I ⊆ k[x 1 , x 2 , . . . , x d ], a problem of great recent interest has been the comparison of the rth ordinary power of I and the mth symbolic power I (m) . This comparison has been undertaken directly via an exploration of which exponents m and r guarantee the subset containment I (m) ⊆ I r and asymptotically via a computation of the resurgence ρ(I), a number for which any m/r > ρ(I) guarantees I (m) ⊆ I r . Recently, a third quantity, the symbolic defect, was introduced; as I t ⊆ I (t) , the symbolic defect is the minimal number of generators required to add to I t in order to get I (t) .We consider these various means of comparison when I is the edge ideal of certain graphs by describing an ideal J for which I (t) = I t + J. When I is the edge ideal of an odd cycle, our description of the structure of I (t) yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.

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Given an ideal I in a Noetherian ring, one can ask the containment question: for which m and r is the symbolic power I (m) contained in the ordinary power I r ? C. Bocci and B. Harbourne study the containment question in a geometric setting, where the ideal I is in a polynomial ring over a field. Like them, we will consider special geometric constructs. In particular, we obtain a complete solution in two extreme cases of ideals of points on a pair of lines in P 2 ; in one case, the number of points on each line is the same, while in the other all the points but one are on one of the lines.

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