In this research paper, we study 3-uniform hypergraphs (Formula presented.) with circular symmetry. Two parameters are considered: the largest size (Formula presented.) of a set (Formula presented.) not containing any edge (Formula presented.), and the maximum number (Formula presented.) of colors in a vertex coloring of (Formula presented.) such that each (Formula presented.) contains two vertices of the same color. The problem considered here is to characterize those (Formula presented.) in which the equality (Formula presented.) holds for every induced subhypergraph (Formula presented.) of (Formula presented.). A well-known objection against (Formula presented.) is where (Formula presented.), termed “monostar”. Steps toward a solution to this approach is to investigate the properties of monostar-free structures. All such (Formula presented.) are completely identified up to 16 vertices, with the aid of a computer. Most of them can be shown to satisfy (Formula presented.), and the few exceptions contain one or both of two specific induced subhypergraphs (Formula presented.), (Formula presented.) on five and six vertices, respectively, both with (Formula presented.) and (Formula presented.). Furthermore, a general conjecture is raised for hypergraphs of prime orders.