AbstractLet (Gi|i∈I) be a family of groups, letFbe a free group, and let$G = F \ast \mathop{\text{\Large $*$}}_{i\in I} G_i,$the free product ofFand all theGi.Let$\mathcal{F}$denote the set of all finitely generated subgroupsHofGwhich have the property that, for eachg∈Gand eachi∈I,$H \cap G_i^{g} = \{1\}.$By the Kurosh Subgroup Theorem, every element of$\mathcal{F}$is a free group. For each free groupH, thereduced rankofH, denotedr(H), is defined as$\max \{\rank(H) -1, 0\} \in \naturals \cup \{\infty\} \subseteq [0,\infty].$To avoid the vacuous case, we make the additional assumption that$\mathcal{F}$contains a non-cyclic group, and we defineWe are interested in precise bounds for$\upp$. In the special case whereIis empty, Hanna Neumann proved that$\upp$∈ [1,2], and conjectured that$\upp$= 1; fifty years later, this interval has not been reduced.With the understanding that ∞/(∞ − 2) is 1, we defineGeneralizing Hanna Neumann's theorem we prove that$\upp \in [\fun, 2\fun]$, and, moreover,$\upp = 2\fun$wheneverGhas 2-torsion. Since$\upp$is finite,$\mathcal{F}$is closed under finite intersections. Generalizing Hanna Neumann's conjecture, we conjecture that$\upp = \fun$wheneverGdoes not have 2-torsion.