Computing and estimating ordered Ramsey numbers

We study ordered Ramsey numbers, which are an analogue of the classical Ramsey numbers for ordered graphs. We first disprove a conjecture with an unexpected graph coloring and improve some already obtained results for a special class of ordered matchings. We expand the classical notion of Ramsey goodness to the ordered case and we attempt to characterize Ramsey good connected ordered graphs. We outline how Ramsey numbers can be obtained computationally and describe a SAT solver based utility developed to achieve this goal, which might be of use to other researchers studying this topic.