We develop a partial-identification approach for English auctions with private values, auction-level heterogeneity unobserved by the analyst, and a potentially unknown number of bidders. Our framework permits restrictions on dependence among bidders’ valuations away from the extreme cases of independent private values and perfect correlation, yielding sharp and informative bounds on the distribution of the highest valuation and the seller’s expected profit. We formulate reserve-price choice as a minimax-regret problem and derive tractable solutions to two relaxations of this problem. We apply our method to a new dataset covering more than 3,500 lots auctioned by Christie’s and Sotheby’s. For modern art sold in New York City with pre-sale estimates between $1 million and $10 million, we find that setting the reserve price 37 percent above the sellers projected value from retaining an unsold lot increases expected profit by approximately $150,000 to $489,000 per lot, equivalent to 27 to 87 percent of average realized profit.