Sperner type theorems with excluded subposets

Let FF be a family of subsets of an nn-element set. Sperner’s theorem says that if there is no inclusion among the members of FF then the largest family under this condition is the one containing all ⌊n2⌋-element subsets. The present paper surveys certain generalizations of this theorem. The maximum size of FF is to be found under the condition that a certain configuration is excluded. The configuration here is always described by inclusions. More formally, let PP be a poset. The maximum size of a family FF which does not contain PP as a (not-necessarily induced) subposet is denoted by La(n,P). The paper is based on a lecture of the author at the Jubilee Conference on Discrete Mathematics [Banasthali University, January 11–13, 2009], but it was somewhat updated in December 2010.

► Nearly largest families of subsets are investigated when certain posets are excluded. ► Certain generalizations of Sperner’s theorem are surveyed. ► Extremal families are considered with excluded subposets.