You can not achieve what you want using POP3 protocol.
POP3 protocol is very limited. In most cases it only returns new messages. When user uses delete command, message is removed and is not available through POP3 again.
Only IMAP offers ability to search, marking/un-marking message as read.
Imap.ExamineInbox selects (opens) INBOX folder in read only mode - most likely you'll not need this method.
Use IMAP protocol (Imap class) use search to find messages and DeleteMessageByUID to delete them.
The code would look more or less like this:
using(Imap imap = new Imap())
{
imap.Connect("imap.server.com"); // or ConnectSSL for SSL
imap.UseBestLogin("user", "password");
imap.SelectInbox();
List<long> uids = imap.Search(
Expression.And(
Expression.HasFlag(Flag.Unseen),
Expression.Before(new DateTime(2014,01,01))
));
imap.DeleteMessageByUID(uids);
imap.Close();
}
Here you can find more information on how to search IMAP server:
https://www.limilabs.com/blog/how-to-search-imap-in-net