  Suppose there is a collection $x_1,x_2,\dots,x_N$ of independent uniform
$[0,1]$ random variables, and a hypergraph $\cF$ of \emph{target structures} on
the vertex set $\{1,\dots,N\}$. We would like to buy a target structure at
small cost, but we do not know all the costs $x_i$ ahead of time. Instead, we
inspect the random variables $x_i$ one at a time, and after each inspection,
choose to either keep the vertex $i$ at cost $x_i$, or reject vertex $i$
forever.
  In the present paper, we consider the case where $\{1,\dots,N\}$ is the
edge-set of some graph, and the target structures are the spanning trees of a
graph, spanning arborescences of a digraph, the paths between a fixed pair of
vertices, perfect matchings, Hamilton cycles or the cliques of some fixed size.
