Van Magnan, University of Montana

Abstract

In extremal set theory, we often ask questions about maximizing the size of a family subject to certain intersecting conditions. The ErdH os-Ko-Rado Theorem describes how the maximum size of any intersecting family is achieved by a 'trivially' intersecting family, in which all members contain a common element. Several generalizations of this problem exist, including the following:

Maximize the quantity $|mathcal{F}|-Delta(mathcal{F})$, where $Delta(mathcal{F})$ is the max degree of an element in the family, such that the family remains intersecting.

The above problem maximizes the textit{diversity} of the family. We define the family's flower base, which combines the methods of delta system and transversal analysis, and show how it can be used to show results in the field. We then partially answer a generalization of the above question, finding extremal constructions maximizing $|mathcal{F}|-Ccdot Delta(mathcal{F})$ for any constant $Cin [0,7/3)$.

See: https://sites.math.rutgers.edu/~kmg326/GCS/GCS.html