Topological Properties of Event Structures

Motivated by the nice labelling problem for event structures, we study the topological properties of the associated graphs. For each n⩾0, we exhibit a graph Gn that cannot occur on an antichain as a subgraph of the graph of an event structure of degree n. The clique complexes of the graphs Gn are disks (n even) and spheres (n odd) in increasing dimensions. We strengthen the result for event structures of degree 3: cycles of length greater than 3 do not occur on antichains as subgraphs. This amounts to saying that the clique complex of the graph of an event structure of degree 3 is acyclic.