Did you have a written contract? Were you given any terms and conditions, perhaps on the reverse of any order form? If so, that may well deal with the situation, but almost certainly it will say that errors in the proof copy which you fail to notice are your responsibility. (Errors introduced by them after you've approved the proof are different, of course.)
Otherwise, was anything said when they sent the proof to you? Was there any suggestion that mistakes could be rectified after you'd approved the proof?
Were you given sufficient time to peruse the proof before you gave the go-ahead? If you'd given them a deadline by which you needed the copies, and they were late getting the proof copy to you, so you had to rush reading the proof, then it's arguable that they're at least partly responsible.
How long is the anthology? The shorter it is, the easier it is for you to have read the proof thoroughly before approving it; conversely, the longer it is, then the less it's reasonable for them to expect you to have checked every single line.
Frankly, though, it's likely they'll say it's down to you, and your best way forward is to go the "more in sorrow than in anger" route, pointing out the mistake and trying to see how you can together achieve a resolution that's good for both of you -- ie negotiate rather than get immediately stroppy with them. Try and get them to do another print run for free, but be prepared to give ground.
In that respect, would you have put more work their way in the future, if it wasn't for this error? If so, use that as a basis for negotiations -- it would be more advantageous for them to do another print run at cost, in the hope of another profitable contract with you, rather than make an enemy of you now, with the risk of bad publicity.
If you can't get another print run you can afford, then perhaps print a few errata slips yourself, if that's feasible.
They were there in the original text I submitted. I have the thumb drive I gave them
