This article proves several graph properties to be elusive. Two of the
main results are 1. If P is a decreasing graph property containing no
graph of girth smaller than 5, then P is elusive. 2. The property of
having matching number at most k, k < left perpendicular\V\/2right per
pendicular, is elusive. The proofs are all based on a topological meth
od developed by Kahn, Saks, and Sturtevant.