Morse Theory and Evasiveness

x0. Introduction. Consider a game played by 2 players, whom we call the hider and the seeker. Let S be a simplex of dimension n, with vertices x 0 , x 1 ; : : :; x n , and M a subcomplex of S, known to both the hider and the seeker. Let be a simplex of S, known only to the hider. The seeker is permitted to ask questions of the sort \Is vertex x i in ?" The seeker's goal is to determine whether is in M, using as few questions as possible. Of course, the seeker is permitted to use the answers to the earlier questions when choosing which vertex to ask about next. We assume that the seeker chooses each question , given the answers to the previous questions, according to a deterministic algorithm, which we call a decision tree algorithm. For any decision tree algorithm A, we denote by Q(; A; M) the number of questions the seeker must ask before it is determined whether or not is in M. The complexity of M, c(M), is deened by c(M) = inf A sup Q(; A; M): M is said to be evasive if c(M) = n + 1, and nonevasive otherwise. That is, M is evasive if for any decision tree algorithm A there is a simplex such that the seeker must ask n + 1 questions before determining if is in M. This decision problem has been the focus of numerous investigations, because it touches upon many subjects, such as complexity theory, graph theory, and topological combina-torics. See, for example, the surveys in Bo] and Bj]. Most of the work has concentrated on the important special case of determining whether or not a graph satisses a monotone graph property when one is permitted to check one edge at a time. The general problem we have presented above seems to have rst appeared in RV1]. In KSS] it was shown that there is a fundamental relationship between the evasiveness of the subcomplex M and its topology.