Stateless Quantum Structures and Extremal Graph Theory

We study hypergraphs which represent finite quantum event structures. We contribute to results of graph theory, regarding bounds on the number of edges, given the number of vertices. We develop a missing one for 3-graphs of girth 4. As an application of the graph-theoretical approach to quantum structures, we show that the smallest orthoalgebra with an empty state space has 10 atoms. Optimized constructions of an orthomodular poset and an orthomodular lattice with no group-valued measures are given. We present also a handcrafted construction of an orthoalgebra with no group-valued measure; it is larger, but its properties can be verified without a computer.