The basic pigeonhole principle says that if more than n items are put into n containers, then some container has at least two items. We typically consider more quantitative versions: how many items do we need if we have 3 containers, and no matter how we distribute the items, at least one container ends with 3 items? (we need at least 7.) Or, how many items do we need if we have 2 containers, and no matter how we distribute the items, either the first container ends up with 3 items, or the second one ends up with 4? (we need at least 6.)

In this talk I consider a version of this principle where we look at infinite linear orders, and consider questions such as: how large should a linear order be, if whenever it is split into two pieces, one of them contains a monotone sequence? (it suffices that the order be infinite.) Or, how large should a linear order be, if whenever it is split into two pieces, either the first piece contains an increasing sequence, or the second one contains a decreasing sequence?