Reflecting Lindel\"of and converging omega_1-sequences

We deal with a conjectured dichotomy for compact Hausdorff spaces: each such space contains a non-trivial converging omega-sequence or a non-trivial converging omega_1-sequence. We establish that this dichotomy holds in a variety of models; these include the Cohen models, the random real models and any model obtained from a model of CH by an iteration of property K posets. In fact in these models every compact Hausdorff space without non-trivial converging omega_1-sequences is first-countable and, in addition, has many aleph_1-sized Lindel\"of subspaces. As a corollary we find that in these models all compact Hausdorff spaces with a small diagonal are metrizable.