A Lower Bound For The Complexity Of Monotone Graph Properties

More than 30 years ago, Karp conjectured that all nontrivial monotone graph properties are evasive, i.e., have decision tree complexity ((n)(2)), where n is the number of vertices. It was proved in 1984 by Kahn, Saks, and Sturtevant [Combinatorica, 4 (1984), pp. 297-306] if n is a prime power by a topological approach. Using their method, we prove a lower bound of 1/3 n(2) - o(n(2)) for general n.