Some New Results on Macaulay Posets

Macaulay posets are posets for which there is an analogue of the classical Kruskal-Katona theorem for finite sets. These posets are of great importance in many branches of combinatorics and have numerous applications. We survey mostly new and also some old results on Macaulay posets. Emphasis is also put on construction of extremal ideals in Macaulay posets