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Abstract:  Proving a 2009 conjecture of Itai Benjamini, we show: For any C there is c > 0 so that for any simple random walk on an n-vertex graph G, the probability that the first Cn steps of the walk see every vertex is less than exp[-cn]. A first ingredient in the proof of this is a similar statement for Markov chains in which all transition probabilities are less than a suitable function of C. Joint with Quentin Dubroff.