Saturation problems in the Ramsey theory of graphs, posets and point sets

Abstract

© 2021 Elsevier Ltd. All rights reserved. In 1964, Erdos, Hajnal and Moon introduced a saturation version of Tur & aacute;n's classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a Kr-free, n-vertex graph with the property that the addition of any further edge yields a copy of Kr. We consider analogues of this problem in other settings. We prove a saturation version of the Erdos-Szekeres theorem about monotone subsequences and saturation versions of some Ramsey-type theorems on graphs and Dilworth-type theorems on posets. We also consider semisaturation problems, wherein we allow the family to have the forbidden configuration, but insist that any addition to the family yields a new copy of the forbidden configuration. In this setting, we prove a semisaturation version of the Erdos-Szekeres theorem on convex k-gons, as well as multiple semisaturation theorems for sequences and posets. (c) 2021 Elsevier Ltd. All rights reserved.